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The evolution of scientific collaboration networks
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Content
THE EVOLUTION OF SCIENTIFIC COLLABORATION NETWORKS
by
Poong Oh
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
In Communication
August 2015
Copyright 2015 Poong Oh
ii
ACKNOWLEDGMENTS
Foremost, I would like to express my sincere gratitude to my advisor, Prof. Peter Monge,
for his continuous support of my PhD study and research, for his excellent guidance,
caring, patience, and providing me with an excellent atmosphere for doing research, and
for pushing me farther than I thought I could go. I could not have imagined having a
better advisor and mentor for my PhD study.
Besides my advisor, I would like to thank the rest of my dissertation committee: Profs.
Janet Fulk and David Kempe, for their encouragement, insightful comments, and hard
questions. I would never have been able to finish my dissertation without the guidance of
my committee members.
I would also like to thank my friends at the Annenberg School: Yasuhito Abe, Shoko
Barnes, Amanda Beacom, Leila Bighash, Joshua Clark, Kevin Driscoll, Martin Hilbert,
Alex Leavitt, Diana Lee, Wenlin Liu, Drew Margolin, Ritesh Mehta, Nina O’Brien,
Katherine Ognyanova, and Mina Park, for all the fun we have had in the last six years.
iii
TABLES OF CONTENTS
ACKNOWLEDGMENTS .................................................................................................. ii
LIST OF TABLES .............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vii
ABSTRACT ..................................................................................................................... xiv
CHAPTER 1: INTRODUCTION ....................................................................................... 1
1.1. Early Research on Social and Communication Networks ............................... 1
1.2. Growth Mechanisms of Real-World Networks ............................................... 7
1.3. A New Approach to Social and Communication Networks .......................... 16
1.4. The Organization of the Dissertation ............................................................. 24
CHAPTER 2: EVOLUTIONARY DYNAMICS ............................................................. 27
2.1. An Overview of Darwinian Evolution ........................................................... 27
2.2. Fundamental Mechanisms of Evolutionary Processes ................................... 30
2.3. The Evolutionary View of Human Rationality .............................................. 47
2.4. The Application of Evolutionary Mechanisms to Network Dynamics .......... 52
CHAPTER 3: THE SELECTION MECHAINSM OF SCIENTIFIC
COLLABORATION NETWORKS ................................................................................. 61
3.1. A Snapshot of the Evolution of Scientific Collaboration Networks .............. 62
3.2. Defining Papers and the Selection Criteria of Journals ................................. 65
3.3. The Impacts of Selection Criteria on Network Structure............................... 70
3.4. The Relationship between Selection Pressure and Network Structure .......... 79
3.5. Summary ........................................................................................................ 93
iv
CHAPTER 4: NETWORK GROWTH BY REPLICATION AND MUTATION ........... 96
4.1. Bipartite Collaboration Networks .................................................................. 98
4.2. A Growth Model of Collaboration Networks .............................................. 104
4.3. Related Works .............................................................................................. 109
4.4. The Decomposition of the Structural Change of Collaboration Networks .. 114
CHAPTER 5: THE EVOUTION OF EMPIRICAL SCIENTIFIC
COLLABORATION NETWORKS ............................................................................... 120
5.1. Data Collection ............................................................................................ 121
5.2. The Growth of Empirical Collaboration Networks ..................................... 123
5.3. Generating Authoring Teams and Possible Networks ................................. 126
5.3. The Structural Change of Empirical Collaboration Networks ..................... 132
5.4. The Estimation of Selection Bias ................................................................. 152
5.5. The Prediction of the Structural Change of Collaboration Networks .......... 158
5.6. Further Analysis with Modified Team Formation Mechanisms .................. 170
CHAPTER 6: DISCUSSION AND CONCLUSIONS ................................................... 180
6.1. Summary of Findings ................................................................................... 180
6.2. Limitations and Directions for Further Research ......................................... 186
6.3. Conclusions .................................................................................................. 189
REFERENCES ............................................................................................................... 191
APPENDICES ................................................................................................................ 213
Appendix 1. The Derivation of the Price Equation............................................. 213
v
LIST OF TABLES
Table 3.1. The relationship between relative area of “sweet spot” and selection
pressure. ..........................................................................................................76
Table 4.1. The parameters for the team formation in collaboration networks .................104
Table 5.1. The eight journals sampled for analysis. ........................................................121
Table 5.2. The changes in the largest eigenvalue λ1 of the collaboration networks of
the eight journals. ..........................................................................................139
Table 5.3. The changes in the spectral gap Δλ of the collaboration networks of the
eight journals. ...............................................................................................146
Table 5.4. The changes in the modularity Q of the collaboration networks of the
eight journals. ...............................................................................................152
Table 5.5. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests
for the percentile ranks of the observed largest eigenvalues λ1. ...................154
Table 5.6. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests
for the percentile ranks of the observed spectral gap Δλ. .............................155
Table 5.7. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests
for the percentile ranks of the observed modularity Q. ................................156
Table 5.8. Regression analyses for the observed change in the largest eigenvalue as
the dependent variable. .................................................................................162
Table 5.9. Regression analyses for the observed change in spectral gap as the
dependent variable. .......................................................................................163
Table 5.10. Regression analyses for the observed change in modularity as the
dependent variable. .......................................................................................164
Table 5.11. Regression analyses for the observed change in the three connectedness
measures of the collaboration network of the Trends in Neurosciences as
the dependent variables (N = 32). .................................................................176
vi
Table 5.12. Regression analyses for the observed change in the normalized spectral
gap of the collaboration network of Trends in Neurosciences as the
dependent variable (N = 32). ........................................................................179
vii
LIST OF FIGURES
Figure 1.1. The schematic depiction of the World Wide Web (taken from Broder et
al., 2000, p. 318). ..............................................................................................8
Figure 1.2. The expected degree distributions of the Erdős-Rényi network (the white
bars) and the scale-free network (the grey bars). ............................................14
Figure 1.3. Random rewiring processes, where p is the proportion of rewired edges,
C is the clustering coefficient of the networks, and L is average path
length (adapted from Watts & Strogatz, 1998, p. 441). ..................................16
Figure 2.1. Population growth by replication (a) under an unlimited supply of
resources and (b) under a limited supply of resources. ..................................34
Figure 2.2. The application of the general model of selection to shopping for apples
(adapted from Price 1995 p. 390). ..................................................................44
Figure 2.3. The Moran process in a finite population. ......................................................45
Figure 2.4. The likelihood that the relative frequencies of subpopulations change in
the absence of mutation and selection. ...........................................................46
Figure 2.5. The vertex copying process, where i is a new node (the grey node), and j
is an existing node (the blue node) chosen at random for the replication
of its linkage pattern (the blue solid lines) with mutation (the red dashed
line). ................................................................................................................58
Figure 3.1. A snapshot of the structural change of the collaboration network of the
Journal of Communication between 2009 and 2010 (the largest
connected components only). .........................................................................63
Figure 3.2. The possible combinations of knowledge elements by single authors (the
red and blue dots) and through collaboration (the green dots). ......................69
Figure 3.3. A journal’s sweet spot defined as the collection of all acceptable papers. .....69
Figure 3.4. Selection of papers (dots) for publications by journals’ selection criteria
(ellipses). .........................................................................................................72
viii
Figure 3.5. The expected forms of dyads in three hypothetical journals. .........................72
Figure 3.6. The possible combinations of knowledge elements by 200 simulated
authors either individually (the white dots) or in teams (the grey dots)
and the sweet spots of three hypothetical journals (the squares). ...................74
Figure 3.7. The relationship between relative area of “sweet spot” and selection
pressure. ..........................................................................................................77
Figure 3.8. Simulated collaboration networks under different levels of selection
pressure. ..........................................................................................................79
Figure 3.9. The numbers of (a) nodes and (b) edges of the 20,000 simulated
collaboration networks. Note: the solid red lines indicate the medians,
and the dashed red line indicates the 95% confidence intervals. ....................81
Figure 3.10. (a) Network density and (b) transitivity of the 20,000 simulated
collaboration networks. Note: the solid red lines indicate the medians
and the dashed red lines indicate the 95% confidence intervals. ....................82
Figure 3.11. Average path length of the 20,000 simulated collaboration networks.
Note: the solid red line indicates the median, and the dashed red lines
indicate the 95% confidence intervals. ...........................................................83
Figure 3.12. The largest eigenvalue as a measure of network robustness against
random failure. Note: Each network has the same numbers of nodes (N =
100) and edges (E = 99). .................................................................................86
Figure 3.13. Spectral gap as a measure of network robustness against “targeted”
attacks. ............................................................................................................88
Figure 3.14. (a) The largest eigenvalue and (b) spectral gap of the 20,000 simulated
collaboration networks. Note: the solid red lines indicate the medians
and the dashed red lines indicate the 95% confidence interval. .....................88
Figure 3.15. A simulated collaboration network under strong selection (the area of
the sweet spot is 1/1000 of the total area). ......................................................89
ix
Figure 3.16. The modularity of the 20,000 simulated collaboration networks (the
largest connected component only). Note: the solid red line indicates the
median and the dashed red lines indicate the 95% confidence interval. ........93
Figure 4.1. A bipartite network of collaborations and its growth. ..................................100
Figure 4.2. The growth of a collaboration network (a) by exact replication, (b) by
replication with replacement of team members with experienced authors,
and (c) by replication with replacement of team members with new
authors. ..........................................................................................................106
Figure 4.3. The decomposition of the structural change of collaboration networks
into (1) the change due to the replication and mutation processes and (2)
the change due to the selection mechanism. .................................................115
Figure 4.4. The patterns in the residuals of the growth model: (a) random and (b)
biased. ...........................................................................................................119
Figure 5.1. The growth of (a) nodes and (b) collaborative connections over time on
the logarithmic scale of the y-axis. ...............................................................124
Figure 5.2. The changes in (a) the transitivity and (b) the relative size of the largest
connected component over time. ..................................................................125
Figure 5.3. The distributions of (a) team size and (b) publication experience and (c)
the weighted and (d) unweighted degree distributions of the eight
collaboration networks on double logarithmic scales. ..................................127
Figure 5.4. Measuring the professional lifetime of an author. ........................................129
Figure 5.5. The number of active authors over time. ......................................................129
Figure 5.6. The average rate of new authors and the relational inertia of the eight
collaboration networks. .................................................................................130
Figure 5.7. (a) The change in the largest eigenvalue λ1 of the collaboration network
of the Journal of Communication between 1952 and 2010 and (b) the
percentile ranks of the observed values of λ1. ...............................................135
x
Figure 5.8. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Philosophy of Science between 1935 and 2010 and (b) the
percentile ranks of the observed values of λ1. ...............................................136
Figure 5.9. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Chemical Reviews between 1925 and 2010 and (b) the
percentile ranks of the observed values of λ1. ...............................................136
Figure 5.10. The observed change in the largest eigenvalue λ1 of the collaboration
network of Trends in Neurosciences between 1979 and 2010 and (b) the
percentile ranks of the observed values of λ1. ...............................................137
Figure 5.11. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Biometrika between 1902 and 2010 and (b) the percentile
ranks of the observed values of λ1. ...............................................................137
Figure 5.12. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of the Journal of Theoretical Biology between 1962 and 2010
and (b) the percentile ranks of the observed values of λ1. ............................138
Figure 5.13. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of the American Journal of Sociology between 1896 and 2010
and (b) the percentile ranks of the observed values of λ1. ............................138
Figure 5.14. The observed change in the largest eigenvalue λ1 of the collaboration
network of Science between 1881 and 2010 and (b) the percentile ranks
of the observed values of λ1. .........................................................................139
Figure 5.15. (a) The observed change in the spectral gap Δλ of the collaboration
network of the Journal of Communication between 1952 and 2010 and
(b) the percentile ranks of the observed values of Δλ. ..................................142
Figure 5.16. (a) The observed change in the spectral gap Δλ of the collaboration
network of Philosophy of Science between 1935 and 2010 and (b) the
percentile ranks of the observed values of Δλ. .............................................142
xi
Figure 5.17. (a) The observed change in the spectral gap Δλ of the collaboration
network of Chemical Reviews between 1925 and 2010 and (b) the
percentile ranks of the observed values of Δλ. .............................................143
Figure 5.18. (a) The observed change in the spectral gap Δλ of the collaboration
network of Trends in Neurosciences between 1979 and 2010 and (b) the
percentile ranks of the observed values of Δλ. .............................................143
Figure 5.19. (a) The observed change in the spectral gap Δλ of the collaboration
network of Biometrika between 1902 and 2010 and (b) the percentile
ranks of the observed values of Δλ. ..............................................................144
Figure 5.20. (a) The observed change in the spectral gap Δλ of the collaboration
network of the Journal of Theoretical Biology between 1962 and 2010
and (b) the percentile ranks of the observed values of Δλ. ...........................144
Figure 5.21. (a) The observed change in the spectral gap Δλ of the collaboration
network of the American Journal of Sociology between 1986 and 2010
and (b) the percentile ranks of the observed values of Δλ. ...........................145
Figure 5.22. (a) The observed change in the spectral gap Δλ of the collaboration
network of Science between 1881 and 2010 and (b) the percentile ranks
of the observed values of Δλ. ........................................................................145
Figure 5.23. (a) The observed change in the modularity Q of the collaboration
network of the Journal of Communication between 1952 and 2010 and
(b) the percentile ranks of the observed values of Q. ...................................148
Figure 5.24. (a) The observed change in the modularity Q of the collaboration
network of Philosophy of Science between 1935 and 2010 and (b) the
percentile ranks of the observed values of Q. ...............................................148
Figure 5.25. (a) The observed change in the modularity Q of the collaboration
network of Chemical Reviews between 1925 and 2010 and (b) the
percentile ranks of the observed values of Q. ...............................................149
xii
Figure 5.26. (a) The observed change in the modularity Q of the collaboration
network of Trends in Neurosciences between 1979 and 2010 and (b) the
percentile ranks of the observed values of Q. ...............................................149
Figure 5.27. (a) The observed change in the modularity Q of the collaboration
network of Biometrika between 1902 and 2010 and (b) the percentile
ranks of the observed values of Q. ................................................................150
Figure 5.28. (a) The observed change in the modularity Q of the collaboration
network of the Journal of Theoretical Biology between 1962 and 2010
and (b) the percentile ranks of the observed values of Q. ............................150
Figure 5.29. (a) The observed change in the modularity Q of the collaboration
network of the American Journal of Sociology between 1896 and 2010
and (b) the percentile ranks of the observed values of Q. ............................151
Figure 5.30. (a) The observed change in the modularity Q of the collaboration
network of Science between 1881 and 2010 and (b) the percentile ranks
of the observed values of Q. .........................................................................151
Figure 5.31. The statistical tests for the estimation of selection bias. .............................153
Figure 5.32. The correlation among the selection biases toward the three network
connectedness measures. Note: the size of circles indicates the size of
collaboration networks in 2010, and the darkness of circles represents the
strength of selection pressures regarding modularity Q. ..............................158
Figure 5.33. The scatter plot of the standardized differences in spectral gap and
modularity between the possible and observed networks with marginal
frequency distributions. ................................................................................167
Figure 5.34. The scatter plot of the standardized differences in modularity and the
largest eigenvalue between the possible and observed networks with
marginal frequency distributions. .................................................................168
Figure 5.35. The scatter plot of the standardized differences in the largest eigenvalue
and spectral gap between the possible and observed networks with
marginal frequency distributions. .................................................................169
xiii
Figure 5.36. The observed and expected changes in the largest eigenvalue λ1 of the
collaboration network of Trends in Neurosciences with the exponential
team sampling. ..............................................................................................172
Figure 5.37. The observed and expected changes in the largest eigenvalue λ1 of the
collaboration network of Trends in Neurosciences with the polynomial
team sampling. ..............................................................................................172
Figure 5.38. The observed and expected changes in the spectral gap Δ𝜆 of the
collaboration network of Trends in Neurosciences with the exponential
team sampling. ..............................................................................................173
Figure 5.39. The observed and expected changes in the spectral gap Δ𝜆 of the
collaboration network of Trends in Neurosciences with the polynomial
team sampling. ..............................................................................................173
Figure 5.40. The observed and expected changes in the modularity Q of the
collaboration network of Trends in Neurosciences with the exponential
team sampling. ..............................................................................................174
Figure 5.41. The observed and expected changes in the modularity Q of the
collaboration network of Trends in Neurosciences with the polynomial
team sampling. ..............................................................................................174
Figure 5.42. The observed and expected changes in the normalized spectral gap ΔλP
of the collaboration network of Trends in Neurosciences with the
uniform team sampling. ................................................................................177
Figure 5.43. The observed and expected changes in the normalized spectral gap ΔλP
of the collaboration network of Trends in Neurosciences with the
exponential team sampling. ..........................................................................178
Figure 5.44. The observed and expected changes in the normalized spectral gap ΔλP
of the collaboration network of Trends in Neurosciences with the
polynomial team sampling. ...........................................................................178
xiv
ABSTRACT
Scientific collaboration networks are a special kind of social and communication
networks in which nodes represents authors, and they are connected in pairs by an edge if
they have worked together to publish a paper. Collaboration networks grow and change
over time as new authors join the networks and new collaborative connections are formed
between authors. However, the addition of nodes and edges is highly selective. Only
authors who can produce acceptable papers can join the network and be connected only
to their coauthors. Therefore, collaboration networks are not just growing but evolving
under the evolutionary constraint that selection prefers particular authors and
collaborative connections.
To account for the structural change of scientific collaboration networks, this
dissertation proposes an evolutionary model that incorporates the three fundamental
mechanisms of evolutionary dynamics—replication, mutation, and selection. The
replication and mutation processes were adopted to formulate the growth mechanism of
collaboration networks. The growth mechanism describes the process in which new
authoring teams are formed by replicating the membership structure of existing teams
with mutation and generates an ensemble of possible networks by adding the new
authoring teams to an ancestral network. Selection takes place at the team level where
particular authoring teams are chosen from all the possible teams based on their
capabilities to produce acceptable papers. The results of numerical experiments suggested
that collaboration networks tend to show high levels of robustness against random
failures which comes at the expense of the tolerance for targeted attacks but to be highly
xv
modular under strong selection, which is consistent with previous literature on network
robustness and modularity.
The proposed model was fitted against empirical collaboration networks. They
were constructed based on the bibliographic information about the papers published in
the eight leading journals in their fields across diverse disciplines, including a
communication journal. The dataset included 168,557 papers and 161,797 authors. The
structural changes of the collaboration networks were measured and decomposed into
two parts: (1) the amount of the change due to the replication and mutation processes and
(2) the amount of the change due to the selection mechanism based on the Price equation.
The decomposition of the observed changes revealed that the structural changes of the
empirical networks were not fully explained by the growth model alone. More
specifically, the observed changes significantly deviated from the changes predicted by
the replication and mutation processes. The analysis of the residuals of the growth model
showed that the observed changes were consistently biased in the direction predicted by
the selection mechanism. Further, a series of regression analyses found that the structural
changes of the empirical collaboration networks were more fully explained when all three
evolutionary mechanisms were included than when they assumed to grow only by the
replication and mutation processes. A set of post-hoc analyses confirmed that the current
findings were replicable when the assumptions of the team formation mechanism were
relaxed, and when structural properties of collaboration networks were measured in
different ways. The major contributions of the dissertation and their implications are
discussed in the final chapter. The limitations of the current study as well as future
research directions are also addressed.
xvi
Keywords: Network evolution, scientific collaboration networks, replication,
mutation, selection, team formation mechanism, network robustness.
1
CHAPTER 1: INTRODUCTION
A network is, in its simplest form, a collection of points connected together in
pairs by lines (Newman, 2010). In the jargon of the field the points are called vertices or
nodes and the lines are called links or edges denoting the interaction among the nodes.
Many objects of interest across disciplines can be thought of as networks, and thinking of
them in this way provides valuable insights into understanding the underlying structure
and mechanisms of the objects.
Networks are classified according to the characteristics of nodes and the nature of
their interaction. When nodes of a network represent individual people, groups,
organizations, or any other kinds of social actors at any levels, and when the links are
their social relations, the network is called a social network (Wasserman & Faust, 1994).
Communication networks are a special kind of social networks, in which the relations are
defined by “the patterns of contact that are created by the flow of messages among
communicators through time and space.” The concept of message here needs to be
understood in a broad sense to refer to “data, information, knowledge, and any other
symbolic forms that can move from one point in a network to another or can be co-
created by network members” (Monge & Contractor, 2003, p. 3).
1.1. Early Research on Social and Communication Networks
Social and communication networks have been the subject of both empirical and
theoretical studies in social science for more than 60 years (Scott, 2000; Wasserman &
Faust, 1994; Watts, 1999), in part because of inherent interest in the emerging patterns of
human interaction, but also because their structure has substantial consequences for social
2
and collective outcomes. In the early 1950s a number of experimental studies were
conducted to examine the effects of communication linkage patterns on group
performance (Bavelas, 1950; Leavitt, 1951; Leavitt & Mueller, 1951). For example, in
the experiment by Bavelas (1950) five-person groups were given a task, which could be
completed when all information distributed among group members was aggregated. By
allowing participants to exchange written messages only with designated others, the
researcher manipulated the communication networks of groups in four different ways,
including circle-, chain-, wheel-, Y-shaped networks. Group performance was measured
by completion time and frequency of errors. The study found that groups with the wheel-
shaped communication network, which was the most centralized, outperformed others,
suggesting that the communication pattern among group members did matter for group
performance.
The structure of social and communication networks was also explored in
observational studies. A well-known early study was conducted by Stanley Milgram
(Milgram, 1967; Travers & Milgram, 1969), which is now better known as the small-
world experiment. In the study, Milgram chose a large sample of participants from a
Nebraska telephone directory and asked them to get a letter to a target person in Boston,
Massachusetts. The instructions were that the letter was to be sent to the target person by
passing it from person to person, but the letter should be passed only to someone whom
the participants knew on a first-name basis. A moderate number of the letters did
eventually reach the target person, and Milgram discovered that the average number of
steps taken from the initial sample to the target person was about six. The finding was
surprising in that it suggested that many pairs of individuals even in a large population
3
were connected by short paths of social relations, so-called, “six degrees of separation”
(Watts, 1999). Despite several methodological issues (e.g., the biases due to the large
proportion of unreturned letters), Milgram’s study is still regarded as one of the most
important and insightful studies of social networks.
An important implication of Milgram’s study, in particular, for communication
research is that it showed that the structure of social and communication networks has
significant impacts on the flow of information through the networks. That is, network
structure determines how fast, widely, and accurately messages can spread in a
population. One of the earliest theoretical models of information flows in networks was
Rapoport’s (1951, 1953a, 1953b, 1954; Rapoport & Rebhun, 1952) rumor spread model.
1
His major contribution was to establish a theoretical relationship between the
connectedness of a network and the degree to which a rumor can spread in the network.
Rapoport’s model was later adopted by epidemiologists and developed into a variety of
disease spread models. Early epidemic models assumed mixed populations, in which
every individual can contact every other with an equal probability and thus can be
infected with a probability proportional to the relative frequency of the infected
individuals in the population (e.g., Daley & Kendall, 1964). However, more recent
models utilize the information about the social network structure of a population to make
more accurate prediction of outbreaks and to suggest more practical preventive
interventions (for a review see Keeling & Eames, 2005). For example, a study by
Christakis and Fowler (2010; see also Garcia-Herranz, Moro, Cebrian, Christakis, &
1
Rapoport used the word “rumor” as a generic term to refer to anything that can be transmitted through
social relations. Actually, he often used the term rumor interchangeably with information and contagious
disease.
4
Fowler, 2014) proposed a practical method for the detection of contagious outbreaks in a
large population using the partial information about its social network structure. Pointing
out the difficulties in mapping the entire network of a large population, the authors
suggested monitoring the “friends” of randomly selected individuals because they are
likely to be central in a network and thus to be infected sooner during the course of an
outbreak.
Social scientists, including communication scholars, have been interested in the
spreading patterns of new ideas or behavior through social and communication networks,
namely, the diffusion of innovations. The earliest study of the diffusion of innovations
was conducted by Coleman, Katz, and Menzel (1957). The authors studied the process in
which physicians in four Midwestern cities adopted a newly introduced drug for their
practices over 15 months. They found that the interpersonal relations of the physicians
and the opinion leaders, who were at the center of the interpersonal network, played
critical roles in the wide spread of the adoption of the new drug. The concept of opinion
leader was originally introduced to account for the importance of interpersonal influences
in political opinion formation and decision making (e.g., voting decision) in comparison
with the effects of political campaigns via mass media (i.e., the two-step flow of
information, E. Katz & Lazarsfeld, 1955). Since the diffusion study by Coleman et al.
(1957), the concept of opinion leader has been commonly operationalized as “central”
nodes in social and communication networks (Rogers, 2005), and various measures of
node centrality have been developed and applied to identify opinion leaders in groups and
communities (Valente, 1995; Wasserman & Faust, 1994). The simplest example of node
centrality is degree centrality, which measures the number of links incident upon a node.
5
Nodes with high degree centrality have more neighbors and thus are likely to be more
influential than nodes with low degree centrality.
On the other hand, Burt (1992) proposed a new approach to understanding the
“central” positions in information and communication networks, based on the classical
Simmelian view of triads (Simmel, 1909; Nooteboom, 2006; see also Krackhardt, 1992)
and, more immediately, on the strength of weak ties argument (Granovetter, 1973). Most
social networks are characterized by dense clusters of strong and redundant connections.
Information within these clusters tends to be homogeneous and redundant, whereas non-
redundant, unique information, for instance, job opportunities in Granovetter’s (1973)
study, is most often obtained through connections to different clusters. Therefore,
individuals with non-redundant connections, that is, those who are connected with others
who are not connected to each other, tend to have greater access to such non-redundant
information and be better off in highly competitive environments. Burt called the
positions connecting different clusters in a network the “structural holes” of the network,
which is what Simmel called tertius gaudens (the rejoicing third). Burt’s theory of
structural holes was later combined with social capital theory (Bourdieu, 1985; Coleman,
1988; Lin, 2001; Portes, 1998; Putnam, 2000), in which social relations are understood as
an important resource in creating benefits for either the individuals or the collective or
both, and has served as a theoretical foundation for understanding a wide range of
economic and strategic behaviors in social networks (Burt, 2007; Jackson, 2008).
Early studies have advanced our understanding of the network structure of social
and communication relations and its implications for social and collective phenomena.
Nevertheless, they suffer from two substantial shortcomings that limit their usefulness:
6
one is theoretical, and the other is methodological. Most previous studies of social
networks before the 1990s treated networks as exogenous structural constraints with a
few exceptions (e.g., Hallinan, 1979; Johnsen, 1986). They focused on the impacts and
consequences of the structural characteristics of networks on some social and collective
outcomes of interest. Put differently, network variables were always placed as causal
variables in the left-hand side of the equation. A common and implicit assumption made
in those studies was that networks are given and static. This assumption excludes the
possibilities that networks themselves change over time, in particular, as the results of
either the internal dynamics among nodes or the external forces surrounding the
networks.
The assumption of static networks in early studies appears to be reasonable, given
the available techniques for data collection and analysis at that time. As Wasserman and
Faust (1994) pointed out, collecting and processing social network data was highly labor-
intensive and error-prone, and the size of the network that could be mapped was therefore
limited, typically, to a few dozens or hundreds of nodes at most. Also, social network
data were often collected by asking people to report on their own interactions. However,
self-reported data were often inaccurate because of imperfect memories and cognitive
biases of respondents in reporting their interactions (Killworth & Bernard, 1976;
Krackhardt, 1987). In addition, it was challenging to specify the boundaries of naturally
occurring social networks (i.e., the set of nodes of a network). In previous studies,
network boundaries were often defined by arbitrary criteria, such as a class of students
(Hallinan, 1979) or an geographical or administrative boundary (Rogers & Kincaid,
1981). However, such boundaries may not fully represent the true social relations among
7
people in question. Poorly defined boundaries often lead to the omission of some
important nodes or edges, whose inclusion might change various network measures
drastically (Kossinets, 2006; Smith & Moody, 2013). For these and many other reasons,
it was difficult to construct even a single complete and reliable social network, and
therefore, most studies focused on cross-sectional networks with limited numbers of
nodes. These methodological and technical limitations of early research have delayed the
development of the theory of network change.
1.2. Growth Mechanisms of Real-World Networks
The increasing availability of digital data since the 1990s (Hilbert & López, 2011)
has resolved many of the methodological and technical issues with traditional social
network research and provided new research opportunities. Among numerous large-scale
network data, the Internet, the network of computers, and the World Wide Web, the
network of web documents, immediately drew scholarly attention (Easley & Kleinberg,
2010; Newman, 2010). Early research on the Internet and the Web was mainly led by
computer scientists, and one of its major goals was to develop efficient search algorithms
that can locate computers and webpages that contain the information requested by clients
(e.g., Brin & Page, 1998; Kleinberg, 1999). For example, a research team led by Broder
(2000) collected over 200 million webpages and identified 1.5 billion hyperlinks among
them. The researchers examined global properties of the Web, focusing on the
reachability of webpages from one another via their hyperlinks. Based on the
reachability, the 200 million webpages were classified into six groups—(1) the pages in
the strongly connected components (SCC), every one of which can reach every other via
hyperlinks, (2) the pages that can reach the SCC but cannot be reached from it (IN), (3)
8
the pages that are accessible from the SCC but do not linked back to it (OUT), (4) the
pages that can neither reach nor be reached from the SCC but are connected to either the
IN or the OUT (tendrils), (5) the pages that can reach from the IN to the OUT without
passing through the SCC (tubes), and (6) all the others that are disconnected to the
previous five groups (disconnected components)—and reconstructed the Web as a “bow
tie” structure (Figure 1.1). Notice that the reachability of one webpage to another implies
that the latter can be discovered by some search algorithm that starts at the former by
exploiting its hyperlinks. Therefore, the bow-tie structure of the Web, more precisely, the
relative sizes of the six different parts provided useful insights for developing efficient
search algorithms on the Web (Easley & Kleinberg, 2010).
Figure 1.1. The schematic depiction of the World Wide Web (taken from Broder et al.,
2000, p. 318).
9
The availability of large-scale network data facilitated another line of research in
the late 1990s, which focused on the growth mechanisms of networks. The interest in the
growth mechanisms developed from the observation of substantial differences in various
network properties between real-world networks and theoretical networks, in particular,
that of Erdős and Rényi’s (1960) random graph model.
Erdős and Rényi (1960) proposed a mathematical model to examine structural
properties of a random network with N nodes and E edges as a function of the increasing
number of edges in the network. Staring with N nodes completely disconnected, a
random graph is generated by selecting a pair of nodes uniformly at random and
connecting them until the number of edges reaches the target number E. Alternatively, a
random graph with N nodes can be generated by placing an edge between each pair of
nodes with an independent probability δ [= E / N (N – 1), i.e., network density] following
a Bernoulli random process. Technically, these two procedures are different, but random
graphs generated by them are “statistically” identical. The number of edges is fixed in the
former, whereas the probability of edges is fixed as network density δ in the latter,
meaning that the number of edges varies. In fact, the network generated by the second
procedure could have no edges at all or have edges between every possible pair of nodes.
By the central limit theorem, however, the expected number of edges is equal to E and
the variance in the number of edges (i.e., sampling variability) converges to zero for large
N. Note that the technical definition of a random network is not in terms of a single
network but in terms of an ensemble, a probability distribution over all possible
networks, as in the definition of a random variable.
10
Erdös and Rényi examined the statistical behavior of structural properties of the
random graph with N nodes and E edges (or a network density of δ). Two structural
properties are closely related to the current discussion—degree distribution and
transitivity. First, the probability Prob (k) that a node has k edges in the Erdös-Rényi
random network, or simply its degree distribution, approaches a Poisson distribution in
the limit of network size approaching infinity (i.e., 𝑁 → ∞):
𝑃𝑟𝑜𝑏 (𝑘 ) =
(𝑁𝛿 )
𝑘 𝑒 −𝑛𝛿
𝑘 !
.
Second, the transitivity of a network is the tendency that two neighbors of a node
are also neighbors of each other, and its measure, the clustering coefficient C, is defined
as the fraction of closed triads of all connected triads (Newman, 2010; Wasserman &
Faust, 1994). In the Erdös-Rényi random network, the probability that pairs of nodes are
connected is constant (i.e., network density δ) and independent of whether the pairs of
nodes share common neighbors. Hence, the clustering coefficient of the random network
is always equal to its density and vanishes as N approaches infinity. For this reason, the
Erdös-Rényi random network shows essentially no transitivity.
The Erdős-Rényi model is one of the best studied models of networks. For the
half century since it was first proposed it has provided a tremendous amount of insight
into the expected structure of networks. The fact that it is both simple to describe and
straightforward to study using analytic methods makes it an excellent tool for
investigating many sorts of network phenomena (Boccaletti, Latora, Moreno, Chavez, &
Hwang, 2006; Gómez-Gardeñes & Moreno, 2006; Newman, 2010). The Erdös-Rényi
model does, however, have some severe shortcomings as a model of empirical networks.
11
There are many ways in which the Erdös-Rényi network is sharply different from real-
world networks,
2
which have been revealed as large-scale empirical networks became
available to study. First, the Poisson distribution does not well fit the degree distributions
of real-world networks, most of which tend to have longer and thicker tails (e.g., power-
law degree distributions). Second, empirical networks in general, and social networks in
particular, are significantly more transitive and clustered than expected by the Erdős-
Rényi model. The observed discrepancy between empirical networks and the Erdős-
Rényi network strongly suggests that the formation of edges among nodes is governed by
some underlying mechanisms other than a Bernoulli random process. A natural question
to ask therefore became “by what mechanism would a network come to have the
observed structural properties?”
As an alternative to the Erdős-Rényi model, Barabási and Albert (1999) proposed
a scale-free network model, one of whose underlying mechanisms is preferential
attachment. The Erdős-Rényi model assumes that the probability of edges among nodes
is independent of the connectivity of those nodes and identical across all pairs of nodes
(commonly referred to as “i.i.d” in statistics). In contrast, the preferential attachment
process assumes that the probability that an edge is formed between a newly arriving
node and an existing node i is proportional to the degree of the existing node, so that the
increase rate of the degree of the existing node i is
𝑑 𝑘 𝑖 𝑑𝑡 =
𝑘 𝑖 ∑𝑘 𝑗 𝑗 , (1.1)
2
For this reason, Newman (2010) pointed out that the exponential random graph models (ERGMs), which
is based upon the Erdős-Rényi model, are “essentially useless models” to investigate the statistical behavior
of empirical networks (p. 582, see also Gómez-Gardeñes & Moreno, 2006).
12
where ki is the degree of node i at a point in time. Put simply, newly arriving nodes
“prefer to be attached” to high-degree nodes over low-degree nodes, and therefore, the
degree of high-degree nodes grows faster than that of low-degree nodes as in the
concentration of wealth in capitalistic societies, so-called “the-rich-get-richer”
phenomena (Hardy, 2010; Simon, 1955).
Further, Barabási and Albert (1999) showed that when the preferential attachment
process is combined with continuous growth of nodes of a network, the degree
distribution of the network follows a power-law:
𝑃𝑟𝑜𝑏 (𝑘 ) ~ 𝑘 −𝛼
That is, the probability 𝑃𝑟𝑜𝑏 (𝑘 ) that a node in the network is connected with k other
nodes is proportional to 𝑘 −𝛼 , where α is a scaling exponent.
Often, a power-law distribution is called a scale-free distribution, and a network
whose degree distribution is a power-law is called a scale-free network. This is because
all power-laws are equivalent up to constant factors, since each is simply a “scaled”
version of the others. To demonstrate, suppose an arbitrary power-law probability
distribution for a quantity x:
𝑃𝑟𝑜𝑏 (𝑥 ) = 𝑐 𝑥 −𝛼 ,
where c is a normalizing constant that ensures ∫𝑃𝑟𝑜𝑏 (𝑥 )𝑑𝑥 = 1 in accordance with
Kolmogorov’s (1950) second axiom of probability stating that the probability that any
elementary event in the entire sample space will occur is 1. Now suppose that the scale or
unit by which the quantity x is measured is increased (or decreased) by a factor of b. The
13
shape of the probability distribution 𝑃𝑟𝑜𝑏 (𝑏𝑥 ) is identical to that of 𝑃𝑟𝑜𝑏 (𝑥 ) expect for
its normalizing constant:
𝑃𝑟𝑜𝑏 (𝑏𝑥 ) = 𝑐 (𝑏 𝑥 )
−𝛼 = 𝑐 𝑏 −𝛼 𝑥 −𝛼 = 𝑏 −𝛼 𝑐 𝑥 −𝛼
= 𝑏 −𝛼 𝑃𝑟𝑜𝑏 (𝑥 ).
Power-law distributions better describe the degree distributions of empirical
networks than Poisson distributions do. Empirical research has shown that many real-
world large-scale networks approximately follow power-law distributions with a scaling
exponent α that varies between 2 and 3 (Clauset, Shalizi, & Newman, 2009;
Mitzenmacher, 2004; Newman, 2005). More specifically, a large proportion of nodes are
poorly connected, whereas a few nodes are highly connected (i.e., hubs), meaning that the
connectivity of nodes in empirical networks is much more heterogeneous than expected
by the Erdős-Rényi model (Figure 1.2). The heterogeneous connectivity of empirical
networks is well captured by the scale-free network model.
However, Barabási and Albert’s (1999) model does not generate any significant
level of transitivity. Their model exclusively focuses on the formation of edges between
new and existing nodes rather than among existing nodes. Because new nodes supposedly
have no neighbors yet, no mutual neighbors are expected between new and existing
nodes. Therefore, the scale-free network is expected to be not transitive at all. Even if
each new node is allowed to form edges to two or more existing nodes, the exiting nodes
that are connected to each other are not necessarily likely to be chosen together. The
choices are strictly based on the degrees of the existing nodes. Therefore, scale-free
networks are no more transitive than the Erdős-Rényi network.
14
Figure 1.2. The expected degree distributions of the Erdős-Rényi network (the white
bars) and the scale-free network (the grey bars).
The small-world network model was proposed by Watts and Strogatz (1998) to
generate the high levels of transitivity of empirical networks. In fact, it is not difficult to
generate highly transitive networks. For example, the regular network shown in Figure
1.3a, in which every node has four edges, can be significantly transitive. In such a
network each node has four neighbors, and there are (
4
2
) = 6 connected triads for each
node, three of which are closed triads. Therefore, the clustering coefficient is .50 = 3 / 6.
A value of .50 is comparable with the clustering coefficients measured for many social
networks (Newman, 2010). Moreover, this value does not depend on the size of the
network N, so it remains the same even when the network size increases or decreases.
While this simple regular network gives a large enough value of the clustering
coefficient, it is clearly unsatisfactory as a model of real-world networks in other
15
respects. One obvious problem is the degree distribution. By definition, all nodes in the
regular network have the same degree, and the degree distribution is uniform, which is
entirely unlike most real-world networks characterized by heterogeneous connectivity.
This problem, however, could quite easily be solved by assigning varying degrees to the
nodes instead of constant ones. A more serious problem is that the average distance
between pairs of nodes in the regular network in Figure 1.3a is much longer than those of
empirical networks or that of the Erdős-Rényi network. To solve the second problem,
Watts and Strogatz (1998) proposed an insightful mechanism of edge formation, namely,
random rewiring.
The small-world model, in its original form, interpolates between a regular
network, such as one in Figure 1.3a, and a random network by rewiring edges from their
original positions to random positions. Starting with the regular network, each of the
exiting edges in the network is chosen with probability p and replaced between two nodes
chosen uniformly at random. This procedure is called random rewiring. The randomly
rewired edges create shortcuts from one part of the network to another and sharply drop
the distance between nodes. As illustrated in Figure 1.3, it takes six steps for node i to
reach node j the farthest one in the regular network (Figure 1.3a), whereas it takes only
three steps after random rewiring (Figure 1.3b). Overall, the average distance between
pairs of nodes decreased from 3.391 to 2.572 by rewiring just 4 of 94 existing edges.
Nevertheless, the network still maintains a high level of transitivity (C = .408). Imagine
that node i represents the initial sample selected in Nebraska and node j is the target
person in Boston as in Milgram’s (1967) study. The fact that letters were delivered from
16
Nebraska to the target person within a few steps strongly suggests the presence of
shortcuts in the social networks of the population.
3
(a) Regular (p = 0) (b) Small-world (0 < p < 1) (c) Random (p = 1)
C = .500
C = .408
C = .136
L = 3.391
L = 2.572
L = 2.424
Figure 1.3. Random rewiring processes, where p is the proportion of rewired edges, C is
the clustering coefficient of the networks, and L is average path length (adapted from
Watts & Strogatz, 1998, p. 441).
1.3. A New Approach to Social and Communication Networks
The introduction of the two influential models of network growth—the scale-free
model and the small-world model—led to further theoretical development in network
3
Note that the presence of shortcuts is a necessary condition for the observation made in Milgram’s study
but not a sufficient condition. Individuals in large-scale social networks do not have access to the
information of the entire network but instead rely upon highly limited local information (e.g., their
immediate neighbors and their neighbors at best), which is not sufficient for them to find the shortest path
to the target person. Successful delivery in a small-world network requires another condition, a high level
of correlation between local structure and long-range connections (i.e., shortcuts), which functions as
critical cues for individuals to find the shortest path. For more details, see Kleinberg (2000a, 2000b).
17
research. A number of “improved” models have been proposed to demonstrate the growth
dynamics of networks and its underlying mechanisms. For example, Krapivsky, Redner,
and Leyvraz (2000) proposed the growing random network (GN) model as a generalized
version of the preferential attachment process. In the GN model attachment kernel [i.e.,
𝜋 (𝑘 ) ~ (𝑘 𝑖 /∑𝑘 𝑗 𝑗 )
𝛼 ] is not necessarily linear (𝛼 = 1) but can be either sub-linear (0 <
𝛼 < 1) or super-linear (𝛼 > 1). Put differently, as new nodes continuously arrive and
form edges to existing nodes, the connectivity of an existing node increases as a function
of its present connectivity, but the function may or may not be strictly linear. The GN
model reproduces a wide range of empirical degree distributions, in particular, those that
deviate from a strict power-law. On one hand, when a preferential attachment with a sub-
linear attachment kernel is assumed (0 < 𝛼 < 1) the GN model predicts a “stretched”
exponential degree distribution. On the other hand, when a super-linear attachment kernel
is assumed (𝛼 > 1), the GN model exhibits a “winner-takes-all” phenomenon, namely,
the emergence of a single dominant node that is linked to almost all other nodes
(Krapivsky et al., 2000, p. 4631).
In addition to the GN model, “even better” models have been proposed. Important
examples include the optimization framework (Papadopoulos, Kitsak, Serrano, Boguñá,
& Krioukov, 2012), the truncated power-law (Mossa, Barthélémy, Stanley, & Amaral,
2002), the size-dependent degree distribution (Dorogovtsev, Mendes, & Samukhin,
2001), the network growth by ranking (Fortunato, Flammini, & Menczer, 2006), the
ultra-small-world network (Cohen & Havlin, 2003), and scaling and percolation in small-
18
world networks (Newman & Watts, 1999) to name a few.
4
A common strategy of these
new models is to add one or more parameters to the previous models and thereby to
increase the flexibility of the models so that they can capture the observed network
properties that the previous models often fails to display.
More importantly, the application of the “improved” models has not been limited
to the “classical” large-scale networks, such as the Internet or the World Wide Web, but
extended to a wide range of real-world networks. Examples include electric power grids
(Albert, Albert, & Nakarado, 2004), transportation systems from airport (Bagler, 2008;
Guimerá & Amaral, 2004; Zhang, Cao, Du, & Cai, 2010), street (Jiang, Yin, & Zhao,
2009), railway (Li & Cai, 2007; Sen et al., 2003), subway (K. Lee, Jung, Park, & Choi,
2008) networks, and networks of galleries (Buhl et al., 2004) and of food ingredients
(Ahn, Ahnert, Bagrow, & Barabási, 2011; Teng, Lin, & Adamic, 2012). More recently,
online social networks, such as Twitter (Cha, Benevenuto, Haddadi, & Gummadi, 2012;
Kwak, Chun, & Moon, 2011; Lerman, Ghosh, & Surachawala, 2012) and Facebook (K.
Lewis, Kaufman, Gonzalez, Wimmer, & Christakis, 2008; Papacharissi, 2009), have
drawn increasing scholarly attention.
In some way many of these networks reflect the features of the social systems in
which the networks are embedded, and therefore, some may loosely be said to be social
and communication networks of the members of the society. However, they do not
directly measure actual contact or exchange of messages between people. For example,
4
In fact, Watts and Strogatz’s (1998) small-world network model has not been elaborated into rigorous
models as much in follow-up studies as Barabási and Albert’s (1999) scale-free network model. This is in
large part because the random rewiring mechanism is difficult to treat by analytical means (Newman,
2010).
19
an increasing number of communication scholars have considered the hyperlinks among
websites as the social and communication relations among the authors/owners of the
websites (Barnett & Sung, 2005; Shumate & Dewitt, 2008; Shumate & Lipp, 2008 etc.).
However, a hyperlink can be created from a website to another without the
authors/owners of the websites knowing each other or without any single piece of
information being exchanged. Technically, a hyperlink is a channel of users, not of
messages, through which the users of a website can move to other websites. In this sense,
hyperlinks among websites are functionally equivalent to airlines among cities, through
which people can travel from one cities to another. Surely, many people travel to develop
and maintain their social relations or to deliver important messages. Nevertheless, airline
networks can hardly be considered as social or communication networks. Likewise,
although hyperlink networks may reflect the social and communication networks of the
website users in some aspect, they are not a good proxy for social and communication
relations among the authors/owners of the websites.
Perhaps, online social networks might be closer to or better reflect actual social
and communication relations among people. Previous studies, however, have found
significant differences in social and communicative behavior between online and offline
contexts. For example, research on computer-mediated communication has shown that
people almost always behave differently when they are situated in online contexts,
regarding impression formation and perception of others (e.g., Ellison, Heino, & Gibbs,
2006; Rosen, Cheever, Cummings, & Felt, 2008) and processing and assessing the
information about others (Donath, 2007; Ramirez, Walther, Burgoon, & Sunnafrank,
2002; Walther, Van Der Heide, Hamel, & Shulman, 2009). In addition, because less
20
structural and normative constraints are imposed, online social relations are more easily
formed than offline or face-to-face relations (Chan & Cheng, 2004; Henderson &
Gilding, 2004; Wang & Wellman, 2010). At the same time, online social relations tend to
be easily dissolved (Noel & Nyhan, 2011; Yue, 2011) and not as functional as offline
relations in terms of gaining social and emotional support (Owen et al., 2010; Trees,
2002) and building trust (Mendoza, Poblete, & Castillo, 2010). Although many
researchers, of course, are interested in them for their own sake, online social networks
do not well represent the social and communication relations among service users.
A more promising source of data is scientific collaboration networks, in which
nodes represent authors, and authors are connected in pairs by an undirected edge if they
have worked together to publish a paper (Newman, 2001c). Scientific collaboration
networks are more truly social and communication networks than other networks
discussed earlier. People who have written a paper together are genuinely acquainted
with one another. More important, they have to communicate with one another to
exchange and share their unique knowledge in order to create new knowledge. In
addition, it is fairly easy to construct reliable large-scale, longitudinal collaboration
networks, because the detailed information about collaborations among authors has been
thoroughly documented and is readily available as in the form of bibliographic
information.
Yet, it is possible to argue that scientific collaboration networks might not fully
reflect the social and communication relations among researchers. In reality, there are
many researchers who know one another very well, communicate with one another quite
frequently, share common research interests, but have never collaborated on the writing
21
of a paper. If that is the case, the absence of collaboration links among researchers might
not necessarily imply that they are not acquainted or communicate with one another.
Therefore, scientific collaboration networks may or may not fully capture the social and
communication relations among researchers In fact, this is not a problem of scientific
collaboration networks. Instead, it is one of the most critical characteristics of scientific
collaboration networks that makes it possible to develop the previous ideas of network
growth into a model of network evolution, which is the primary goal of this dissertation.
Suppose that a collaboration network is constructed based on the bibliographic
information of all the papers published in a particular journal. In that case, the absence of
collaborative connections between researchers indicates that none of their papers have
been submitted or accepted for publication in the journal but does not necessarily mean
that they have never collaborated. They might have worked together, perhaps more than
once, but they simply failed to produce an acceptable paper, which is quite common in
reality. It is also possible that they have really never collaborated. In this case, their social
and communication relations were not meaningful enough to start a research project.
Therefore, edges in scientific collaboration networks can be seen as successful
collaborations in producing acceptable papers. Similarly, nodes are authors who can
contribute to producing acceptable papers. In other words, nodes and edges observed in
scientific collaboration networks are selected ones out of many other possible nodes and
edges rather than any ones of them. Further, a collaboration network observed at a given
moment is just one out of many possible networks into which it would have changed if
different sets of nodes and edges were added.
22
These refined definitions of nodes and edges contain important implications for
understanding the growth dynamics of scientific collaboration networks. Unlike Barabási
and Albert’s (1999) scale-free network model, in which a network grows as any new
nodes arrive, scientific collaboration networks grow only when authors who can produce
acceptable papers join the networks. Also, unlike Watts and Strogatz’s (1998) small-
world network model, in which existing edges can be replaced between any pairs of
nodes, collaboration links are formed only between authors whose collaboration can
produce acceptable papers. This suggests the possibility that some substantial selection
mechanism operates on both the addition of nodes and the formation of edges during the
growth of scientific collaboration networks. In short, scientific collaboration networks are
not just growing but evolving over time.
The term evolution is colloquially understood as the gradual change or
development of something over time, especially from a simple to a more complex form
(Bowler, 1975). For example, Erdős and Rényi (1960) used the term graph evolution to
refer to as the “gradual development and step-by-step unraveling of the complex structure
of graph, when the number of edges increases, while the number of nodes is a given large
number” (p. 19 emphasis added). Similarly, Barabási and Albert (1999) used the term
network evolution to indicate the structural change that resulted from “the addition (and
sometimes removal) of new nodes and connections between nodes” (p. 511). Of course,
evolution is a kind of change, but not all changes should be considered as evolution. For
instance, as a car gets older, its appearance changes over time: it rusts, its parts get worn,
and so on. The car’s appearance has definitely changed but would not be said to have
evolved.
23
Evolution occurs strictly through three fundamental mechanisms—replication,
mutation, and selection, which differentiate evolution from other kinds of change
(McElreath & Boyd, 2007; Nowak, 2006). In brief, replication occurs when individuals
in a population create copies of themselves, leading to the growth of the population.
Mutation occurs when individuals fail to make identical copies of themselves, providing
new variation in the population. Finally, selection occurs when individuals with certain
traits are preferred and chosen for further replication, resulting in different growth rates
among individuals with different traits. These three mechanisms serve as the basic
building blocks of evolutionary theory.
Many natural and artificial complex systems can be better understood when they
are thought of as networks that are simply composed of points and lines. Likewise, the
dynamic changes of complex systems can be better understood if they are properly
formulated as the three fundamental mechanisms of evolution. With this perspective in
mind, this dissertation aims to provide a new insight into understanding the dynamic
growth and change of scientific collaboration networks, more generally, social and
communication networks, from the evolutionary point of view. More specifically, the
process by which academic papers are written, submitted, and finally accepted by
journals will be described by the selection mechanism that operates as an external force at
the team level (Chapter 3). The process by which scientific researchers work on papers in
teams and form collaborative connections among the team members will be formulated as
the replication with mutation that operates at the local level (Chapter 4). By integrating
these three processes, a theoretical model will be proposed as one plausible explanation
for the structural change of collaboration networks and tested against empirical data.
24
1.4. The Organization of the Dissertation
The rest of this dissertation is organized as follows. Chapter 2 discusses major
concepts of Darwinian evolutionary theory, which will serve as the theoretical basis of
the present model of network evolution. The first section briefly reviews the early
development of Darwinian evolution in comparison with other evolutionary thoughts.
The second section outlines the three fundamental processes of evolutionary dynamics—
replication, mutation, and selection—and other relevant concepts. The third section
discusses the applicability of biological evolutionary theory to human-involved complex
systems in which humans play controlling roles. In particular, it discusses the roles that
human rationality plays in evolutionary processes based on Maynard Smith’s (1982)
evolutionary game theory. The final section revisits three previous network models from
a viewpoint of evolutionary theory, including Barabási and Albert’s (1999) scale-free
network model, Watts and Strogatz’s (1998) small-world network model, and Kleinberg
et al.’s (1999) vertex copying model.
Chapter 3 develops a model of the selection mechanism in the course of evolution
of scientific collaboration networks based on Price’s (1995) general model of selection.
First, the process by which academic papers are written, submitted, and finally accepted
by journals is modeled based on the stochastic process proposed by Gilbert (1997) and
elaborated by Guimerà, et al. (2005). Based on the stochastic model, a series of numerical
experiments was carried out to investigate the impacts of the selection of papers on the
structure of collaboration networks. The simulation results suggested that collaboration
networks tend to be better connected, in terms of cost-efficiency, under strong selection,
which is consistent with previous literature on network robustness and modularity.
25
Chapter 4 proposes a growth model of collaboration networks through replication
and mutation at the local level based on the self-organization model of bipartite networks
(Ramasco, Dorogovtsev, & Pastor-Satorras, 2004; Ramasco & Morris, 2006) and the
vertex copying process (Kleinberg et al., 1999; Kumar et al., 2000; Kumar, Raghavan,
Rajagopalan, & Tomkins, 1999). The proposed model describes the mechanism by which
groups of authors form teams to publish papers, which are called “authoring teams.” An
authoring team is formed simply by replication, when an existing team repeat their
collaboration to produce another paper. However, mutation can occur when the
incumbent members of the team are replaced by new members. An authoring team is the
smallest unit that can possibly (but not necessarily) alter the structure of a collaboration
network when the team is added to the network. Depending on which team is added, a
network can change in many different ways. Therefore, the team formation mechanism
(1) generates possible authoring teams by the replication of existing teams with mutations
and thereby (2) produces networks into which a current network possibly grows when the
generated teams are added.
The structure of collaboration networks changes over time due to either the
growth mechanism (i.e., the replication with mutation in team formation) or the selection
mechanism or most likely both. Chapter 5 examines empirical collaboration networks and
analyzes the observed change in those networks to identify the sources of the change. The
collaboration networks were constructed based on the bibliographic information of the
papers published in eight leading journals across different disciplines, including (1) the
Journal of Communication (social science), (2) Philosophy of Science (arts and
humanities), (3) Chemical Reviews (chemistry), (4) Trends in Neurosciences (medical
26
science), (5) Biometrika (mathematics), (6) the Journal of Theoretical Biology (biology),
(7) the American Journal of Sociology (social science), and (8) Science
(multidisciplinary).
Finally, Chapter 6 summarizes the findings of the current study and discusses the
implications and limitations of the current model of the evolution of collaboration
networks. Also, it suggests the directions for future research.
27
CHAPTER 2: EVOLUTIONARY DYNAMICS
The original formulation of evolutionary theory and many of the investigations
during its first hundred years dealt with the origin and adaptation of species through the
transmission of genetic information from one generation to another. But what is
transmitted through evolution does not have to be genetic information. Wherever there is
replication with mutation under selection in any forms, there is evolution (Dawkins,
1976). Therefore, the application of evolution theory does not have to be limited to
genetic or biological evolution. Indeed, evolution theory itself has evolved into a
powerful theoretical framework for the study of complex, adaptive, dynamic systems and
applied to a wide range of non-biological phenomena (Bowler, 1975; Kauffman, 1993;
Monge, Heiss, & Margolin, 2008; Nelson & Winter, 1982; Skyrms, 2000). The next two
chapters develop the evolutionary model of scientific collaboration networks by utilizing
the theoretical framework of evolutionary theory. Before proceeding, this chapter outlines
the basic ideas and key concepts of evolutionary theory, which will be used in the
following chapters.
2.1. An Overview of Darwinian Evolution
On the 24th of November in 1859 the first edition of the book On the Origin of
Species by Means of Natural Selection was published and immediately became one of the
most important and influential books in human intellectual history. The book presents a
body of evidence that species are generally not static in their forms but instead change
over generations. The author Charles Darwin (1809-1882) formulated this change
mechanism and called it “evolution.” However, the concept of evolution was not invented
by Darwin. The idea that species change over time was already introduced by the ancient
28
Greek philosopher, Lucretius (99 BC.-55 BC.) in his book De rerum natura (On the
Nature of Things) (G. Campbell, 2003). In addition, the term evolution had been
commonly used in biology, in particular, in embryology to describe the development of
an embryo from the fertilization of the ovum to the fetus stage even before Darwin was
born (Bowler, 1975).
However, there is a fundamental difference between Darwinian evolution and
other evolutionary thoughts, which allows it to be an extremely powerful and general
theoretical tool for understanding many kinds of change. A brief discussion of the
difference would provide the context for understanding the logics that underlies
Darwinian evolution, in particular, the concept of natural selection. Consider Lamarckian
evolution, which was a dominant idea of evolution before Darwin. It was proposed by the
French scientist Jean-Baptiste Lamarck (1744-1829) in his Floreal lecture of 1800. An
illustrative application of Lamarckian evolution is the account for the process by which
the ancestral giraffes have evolved into modern giraffes with long necks and legs
(Agutter & Wheatley, 2007; Beatty, 1984). Lamarck believed that the ancestors of
giraffes used to have short necks and legs. However, when there was not enough food
that they could reach, they stretched their necks and legs to reach more food during their
lifetimes. Their offspring inherited the acquired modification and thereby could have
longer necks and legs as juveniles than their parents had had as juveniles.
5
The offspring
further stretched their necks and legs to reach more food, and their offspring had even
5
In modern biology, it is known that traits acquired during over the life of an organism (e.g., the stretched
neck in the giraffe example) are not transmitted to its offspring (Agar, Drummond, Tiegs, & Gunson, 1954;
Crew, 1936). However, the heritability of acquired traits (also called soft inheritance) could be possible, in
particular, through social learning and imitation in the evolution of non-biological systems (Boyd &
Richerson, 1982; Jablonka, Lamb, & Avital, 1998; Usher & Evans, 1996).
29
longer necks and legs than their predecessors. As the results of the repetition of such a
process over generations, all the descendant eventually have long necks and legs. The
important point here is how the variations that help the survival of the species arose in the
very first place (Beatty, 1984; Koonin & Wolf, 2009; Mesoudi, Whiten, & Laland, 2004).
According to the Lamarckian account, variations occur in response to the survival needs
of species. More specifically, the ancestral giraffes, or any species in general, should be
able (1) to collect and process information about their environment, (2) to find possible
ways to increase their survival chance, and finally (3) to select one out of the possible
ways such that their survival chances are maximally increased in the given environment.
In short, species themselves are the key players in the course of evolution.
Let us consider an alternative scenario. What would happen if variations occur
“by accident,” that is, if, among the ancestors of giraffes, a few slightly longer-necked
and legged offspring happen to be born? In that case, of course, it would be reasonable to
expect that the offspring with even-shorter necks and legs would be born as likely as
those with longer necks and legs. Those with longer necks and legs would be able to
reach more food and thus outbreed the others, passing on their traits to their offspring. As
the results, individuals with longer necks and legs would be a bit more common in the
next generation than in the previous generation. The repetition of such a process over
generations would eventually result in the dominance of the long-necked and long-legged
individuals in the giraffe population. This alternative scenario suggests that purely
accidental or “random” variations are sufficient to drive the evolution of shorter-necked
grazers into modern giraffes (Beatty, 1984; Koonin & Wolf, 2009). Such accidental
variations do not require individual animals to have the ability to make right decisions for
30
their survival. Instead, the decision on the direction of evolution is made by the given
environment, more specifically, by the location and distribution of leaves in the giraffe
example. In this alternative scenario, nature is the key player in the course of evolution as
the agent of selection. This is the core idea of Darwinian evolution and why it is called
“evolution by means of natural selection.”
Nature is, in general, conceived to play the key role in the evolutionary processes
as the agent of selection. But selection can be made by other agents. Notice that Darwin
developed his idea of natural selection to explain the biological data that he had collected
from natural environments, such as the Galapagos archipelago. But he himself pointed
out that selection can occur in artificial environments too, in which nature is no longer
the agent of selection. His examples were the evolution of fancy pigeon and ancon sheep
under domestication (Darwin, 1859). Neither pigeons nor sheep purposively develop
certain traits, such as long beak or short leg, to outbreed others. Instead, human breeders
select types of animals to breed more than others, playing the role of the agent of
selection under domestication. A common feature of the two evolutionary processes
under natural selection and under domestication is that variation is always random and
blind to survival, whereas selection is decidedly non-random.
2.2. Fundamental Mechanisms of Evolutionary Processes
Evolutionary theory is one of the most, perhaps the only, widely applied scientific
theory beyond its founding discipline. Its wide application is largely due to its two unique
characteristics. First, evolutionary theory is, in essence, a theory of change, which is one
of the most common topics across disciplines from physics (e.g., Callebaut & Rasskin-
Gutman, 2005; Gros, 2009) to economics (e.g., Nelson & Winter, 1982). Second and
31
more important, evolutionary dynamics is composed of rather simple mechanisms—
replication, mutation, and selection—each of which does not require any context-specific
assumptions (McElreath & Boyd, 2007). Because these three mechanisms are well
grounded on precise mathematical foundations, the key concepts of evolutionary theory
and their relationships are easily translated and applied to many kinds of change
(Cressman, 2003; Nowak, 2006). This section reviews the three mechanisms and other
relevant concepts based on the key literature in modern evolutionary theory.
2.2.1. Replication
Replication (also called self-replication) is referred to as the behavior of a
dynamical system that autonomously manufactures identical copies of itself using raw
materials taken from its environment (von Neuman, 1966). A typical example is asexual
single-cell organisms, which reproduce by cell division. During cell division, their
genetic information is passed on to their offspring. When replication is assumed, there
will be a group of individuals who share common traits, and such a group is called a
population (Maynard Smith, 1974). As replication is continued, a population grows in
size. Under no environmental constraints (e.g., an unlimited supply of resources) a simple
replication mechanism results in exponential population growth (Malthus, 1798;
McElreath & Boyd, 2007; Nowak, 2006).
To illustrate, consider a population of single-cell organisms that reproduce by cell
division. Let 𝑥 (𝑡 ) be the abundance of cells at time t. Further, suppose that cells divide at
rate r, which is their intrinsic reproduction rate. Here, “at rate r” means that each cell is
expected to divide r times per unit of time. Put differently, a parent cell divides into two
daughter cells every 1/𝑟 units of time on average, and it takes 1/𝑟 units of time for each
32
cell to get ready for another round of cell division. That is, each cell has to wait for 1/𝑟
units of time to start over the cell division process again. For this reason, the inverse of
reproduction rate 1/𝑟 is called waiting time, cell cycle, or generation (Nowak, 2006). The
growth of the cell population over time is described by a differential equation:
𝑑𝑥 𝑑𝑡 = 𝑟𝑥 .
(2.1)
This equation has the explicit solution that can be found using traditional methods
of integration, as is done below. By separating the variables y and t, the equation can be
rewritten as
(
1
𝑥 )𝑑𝑥 = 𝑟𝑑𝑡 .
This results in the simplified integration problem,
∫(
1
𝑥 )𝑑𝑥 = ∫𝑟𝑑𝑡 .
The intermediate steps leading to a solution for this problem are
ln|𝑥 | = 𝑟𝑡 + 𝐶 ,
where C is a constant of integration.
𝑒 ln|𝑥 |
= 𝑒 (𝑟𝑡 +𝐶 )
= 𝑒 𝑟𝑡
∙ 𝑒 𝐶 ,
𝑥 = ±𝑒 𝐶 ∙ 𝑒 𝑟𝑡
.
If the abundance of cells at time 0 is given as 𝑥 0
(i.e., the initial population size), the
solution for the above differential equation is found as an exponential function (Figure
2.1a).
33
𝑥 (𝑡 ) = 𝑥 0
𝑒 𝑟𝑡
. (2.2)
However, exponential population growth is unrealistic. No self-replicating
populations can grow exponentially because real environments can supply only a limited
amount of resources. Thus, real populations will stop growing when they reach the
maximal size that their environments can sustain. In evolutionary ecology, the maximal
size of a population is called the carrying capacity of the environment (Maynard Smith,
1974; see also Monge et al., 2008). When the carrying capacity of an environment U is
considered, the growth of a population is described as follows:
𝑑𝑥 𝑑𝑡 = 𝑟𝑥 (1 −
𝑥 𝑈 ).
(2.3)
As long as 𝑥 < 𝑈 , the increase rate 𝑑𝑥 /𝑑𝑡 is positive, meaning that the population keeps
growing. But when the population size reaches the carrying capacity (i.e., 𝑥 = 𝑈 ), the
increase rate becomes zero, meaning that the population stops growing. The solution for
the above differential equation is found as a logistic function (Figure 2.1b):
𝑥 (𝑡 ) =
𝑈 𝑥 0
𝑒 𝑟𝑡
𝑈 + 𝑥 0
(𝑒 𝑟𝑡
− 1)
,
(2.4)
where 𝑥 0
is the initial population size as before. Notice that exponential population
growth is a special case of logistic growth with an unlimited carrying capacity. When U
approaches infinity (𝑈 → ∞), the equations (2.3) and (2.4) are reduced into (2.1) and
(2.2), respectively.
34
(a) Exponential growth
(b) Logistic growth
Figure 2.1. Population growth by replication (a) under an unlimited supply of resources
and (b) under a limited supply of resources.
Population growth becomes more dynamic and complex when two or more
populations reside in a common environment. When multiple populations rely upon the
same kind of resources, they will compete for the resources. As a result, the growth of
one population leads to the decline of other populations as in a zero-sum game. On the
other hand, if different populations reply upon different kinds of resources, they do not
have to compete but may instead coexist in a common environment. Another interesting
and more complex population growth occurs when one population is the prey of the other
population (e.g., hares vs. wolves). The growth of the prey population leads to an
abundance of food for the predator population and thus the growth of the predators.
However, the excessive growth of the predators will result in the near-extinction of the
prey population, which in turn leads to the decline of the predators. Again, the decline of
the predator population will lead to the growth of the prey population, and so on. These
35
inter-population dynamics and other possibilities are well formulated by the Lotka-
Volterra model.
The following system of differential equations is a generic form of the Lotka-
Volterra model for two populations X and Y, whose sizes are denoted by x and y,
respectively:
{
𝑑𝑥 𝑑𝑡 = 𝛼𝑥 − 𝛽𝑥𝑦 𝑑𝑦 𝑑𝑡 = 𝛾𝑦 − 𝛿𝑥𝑦 ,
(2.5)
where 𝛼 and 𝛿 are the intrinsic reproduction rates of populations X and Y, respectively,
while 𝛽 is the influence of the size of population Y on the growth of population X, and 𝛾
is the influence of the size of population X on the growth of population Y. The signs of 𝛽
and 𝛿 define the nature of the relationship between the two populations. For example,
when both 𝛽 and 𝛿 are positive, the growth of one population leads to the decline of the
other. Thus, their relationship is competitive. When both 𝛽 and 𝛿 are negative, the growth
of one population leads to the growth of the other, implying that their relationship is
symbiotic.
6
Finally, when 𝛽 is positive and 𝛿 is negative, population X is the prey of
population Y.
The self-replication mechanism is the first and the most essential mechanism in
Darwinian evolution without which any change over time should not be seen as evolution
(McElreath & Boyd, 2007). For this reason, the modern evolutionary theorist Martin
Nowak said, “Strictly speaking, neither genes, nor cells, nor organisms, nor ideas evolve.
6
Symbiotic relationships can be further sub-categorized as mutualism, commensalism, amensalism, and
parasitism according to the dependence of one population on the other. For more details see Campbell and
Reece (2002).
36
Only self-replicating populations can evolve. […] Thus, populations are the fundamental
basis of any evolution” (2006, p. 26).
2.2.2. Mutation
The replication mechanism alone generates interesting dynamics and complexities
of population growth. However, these dynamics and complexities are insufficient to fully
understand evolutionary processes. If only the replication mechanism operated since the
very first living organism came into existence on this planet billions of years ago, the
variety of species should have remained the same, which cannot explain the biological
diversity that Darwin witnessed in the Galapagos archipelago. This is the reason that
Darwin conceived of an opposing mechanism, specifically, the errors in the process of
replication, called mutation. A replication mechanism is not always perfect. Individuals
in populations often fail to make identical copies of themselves. Such mistakes generate
variants of the original population and increase the diversity in the population (Nowak,
2006).
The terms “mutant” and “mutation” are commonly used to describe something
undesirable or defective. This is because individuals with mutant traits are, in general,
greatly subjected to the danger of elimination during evolutionary processes (Allen,
Traulsen, Tarnita, & Nowak, 2012). However, mutation plays a critical role at the
population level by increasing the trait diversity within populations, which serves as a
way for populations to adapt to changing environments (Nowak, 2006). With more
variation, it is more likely that some individuals in a population will possess traits suited
for new environments. Those individual are likely to survive and reproduce. As a result,
the population can continue for the next generation. In this sense, evolving populations
37
should maintain some level of trait diversity through mutation to avoid extinction due to
sudden environmental changes. In short, “Life takes advantage of mistakes” (Nowak,
2006, p. 33).
The maintenance of trait diversity through mutation is critical for the survival and
adaptation of non-biological systems, too. An excellent example is March’s (1991) study
on the adaptive strategies in organizational learning. From the view point of evolutionary
economics, the author examined two different types of adaptive strategies: the
exploration of new possibilities and the exploitation of old certainties. The examples of
exploration include search, variation, risk-taking, experimentation, and innovation. Those
of exploitation includes refinement, choice, reproduction, execution, and mutual learning
between organizational members. Exploration conceptually corresponds to mutation and
exploitation to replication. His theoretical analysis found that organizations that engage in
exploitation to the exclusion of exploration are likely to find themselves trapped in
suboptimal stable equilibria. As a result, maintaining an appropriate balance between
exploration and exploitation was a primary factor in the survival and prosperity of
organizations.
From the Darwinian perspective, mutation occurs purely by chance. Individuals in
evolving populations do not make “mistakes” intentionally or purposively (Koonin &
Wolf, 2009; see also D. Campbell, 1965), as opposed to the Lamarckian account.
However, it is possible to raise questions about whether mutation is pure randomness, in
particular, in non-biological evolution. Variants in non-biological evolution can arise by
accident and some of them eventually become dominant in populations. Examples are
many ground-breaking technological innovations that were invented by accident (Basalla,
38
1988). However, many other innovations are the products of well-designed and pre-
planned projects rather than chance. In those cases, the Darwinian view of random
mutation may not be fully applicable to non-biological evolution. In fact, there is an
ongoing controversy over the purposiveness of human behavior based on rationality in
evolutionary processes. Section 2.3 will discuss this issue more fully from the
evolutionary game-theoretic view.
2.2.3. Selection
As a result of mutation a population is further divided into subpopulations, each
of which has distinctive traits from one another. If the difference in their traits is critical
for each to survive (e.g., the length of neck in the giraffe example), some subpopulation
will outbreed the others, leading to the increase in its proportion in the population of the
next generations. In the language of population growth, subpopulations with different
traits will grow at different rates. The faster-growing subpopulations are fitter than the
others in the given environment, and thus, more likely to be selected for reproduction by
nature. As a corollary, the difference in growth rate among subpopulations indicates the
presence of natural selection, and further, the variance in growth rate indicates the
strength of natural selection pressure, which is known as Fisher’s (1930) fundamental
theorem of natural selection: “The rate of increase in fitness of any organism at any time
is equal to its genetic variance in fitness at that time” (p. 50).
Fisher’s theorem can be better understood when we consider the logics of random
experimental designs and the F-test he invented (Fisher, 1925). Suppose that a number of
patients who suffer from an illness are randomly assigned into two groups. A placebo is
given to a group (i.e., the control group), while a newly invented medicine is given to
39
another group (i.e., the experimental group). In the language of evolution, the patient
population is divided into two subpopulations by the type of treatments they receive.
Further suppose that 10% of patients in the control group and 50% of patients in the
experimental group recovered in a week. In that case, the difference, more precisely, the
variance in recovery rate between the two groups indicates the effect of the new
medicine. The more effective the new medicine is, the larger the variance would be
observed. Further, the significance of the effect can be tested using the F-test. In a similar
manner, the variance in growth rate indicates the degree to which nature or other agents
of selection have played out in population growth.
However, Fisher’s theorem had been often misunderstood, being read as saying
that the average fitness (growth rate) of a population would always increase (Frank &
Slatkin, 1992). A monotonous increase in the average fitness means that natural selection
takes out all individuals that are less suited for their environments, eventually leading to
the survival of the fittest
7
, which is one of the most common misconceptions about
natural selection (Gregory, 2009). George Price clarified Fisher’s original statement by
reformulating it into a deterministic covariance equation, known as the Price equation (G.
Price, 1970; see also Frank, 2012). The Price equation (1) formulates evolution as the
change in some heritable trait of a population over generations, (2) associates the value of
the trait with the growth rate of the subpopulation characterized by that trait value, and
7
Although the “survival of the fittest” is a common way to describe natural selection, the “survival of the
fit enough” is more accurate. According to Kauffman (1993), an evolving population adapts to its
environment by adjusting various characteristics of itself through natural selection. But the adjustment does
not always increase its overall fitness, because changing one characteristic can lead to the increase in the
fitness associated with the changed characteristic but may decrease the fitness related to other
characteristics at the same time. Thus, the population is not always optimized by natural selection but tends
to remain in a suboptimal state “despite natural selection” (p. 35, emphasis in original).
40
(3) evaluates the effects of the selection and the trait transmission (by replication and
mutation) on the change in the overall characteristic of the population from one
generation to the next. The Price equation makes it possible to infer the direction and
intensity of natural selection from the observed change in certain trait in a population.
To demonstrate, suppose a population of n individuals, each of which has a
heritable trait described by a value of z. In the giraffe example, each individual can be
characterized by their neck length. Then, the population can be grouped by the trait (e.g.,
those with the 10-inch-long neck, those with the 11-inch-long neck, and so on). In
principle, there could be as few as just one group of all the individuals, if they are of an
equal trait, and as many as n groups of one individual each, if every individual is
different. Let subscript i identify the group with trait 𝑧 𝑖 and let 𝑤 𝑖 be the growth rate of
that group. The Price equation states that the change in the average trait of the population
from one generation to the next, denoted by Δ𝑧 ̅ (= 𝑧 ̅
′
− 𝑧 ̅ ), is decomposed into two parts:
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 Δ𝑧 𝑖 )
𝑤̅
,
(2.6)
where ∆𝑧 𝑖 is the change in trait value of the offspring of group i and defined as 𝑧 𝑖 ′
− 𝑧 𝑖 .
The first part, the covariance between growth rate and trait value 𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )/𝑤̅ gives the
amount of change due to the selection. The second part 𝐸 (𝑤 𝑖 Δ𝑧 𝑖 )/𝑤̅ indicates the
amount of change due to the trait transmission (see Appendix 1 for the proof).
When the trait 𝑧 𝑖 of group i does not change from the parent to child generations
(i.e., no mutation), that is, ∆𝑧 𝑖 = 𝑧 𝑖 ′
− 𝑧 𝑖 = 0, the second term in the Price equation
becomes zero resulting in a simple version of the Price equation:
41
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 ∙ 0)
𝑤̅
=
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
.
(2.7)
This simplified Price equation states that the average value of a heritable trait in the child
population will change as the results of selection to the extent to which the value of the
trait is associated with the growth rate of the subpopulation of individuals characterized
by the value, 𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 ). In short, the covariance term indicates the direction and
intensity of natural selection.
When the growth rate of subpopulations itself is the trait of interest (i.e., 𝑤 𝑖 = 𝑧 𝑖 ),
the Price equation is further simplified as
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑤 𝑖 )
𝑤̅
=
𝑣𝑎𝑟 (𝑤 𝑖 )
𝑤̅
,
(2.8)
where 𝑣𝑎𝑟 (𝑤 𝑖 ) is the variance in growth rate among subpopulations. The above equation
states that the intensity of selection pressure Δ𝑧 ̅ is always equal to the variance in growth
rate among subpopulations 𝑣𝑎𝑟 (𝑤 𝑖 )/𝑤̅, as Fisher (1930) originally stated.
A more important contribution of Price was that he proposed a general model of
selection that unifies all forms of selection, from natural selection in biological evolution
to shopping at a store, voting for political candidates, and choosing paintings from an
exhibition (G. Price, 1995, written circa 1971 published posthumously). Interestingly, he
developed the model in the way that Claude Shannon (1948) developed the mathematical
theory of communication. Price pointed out that the single most important prerequisite of
Shannon’s theory was the definition of information as a measurable physical entity,
which omitted all considerations of meaningfulness of messages but focused on attributes
relevant to the design of efficient communication systems (see also Pierce, 1961).
42
Similarly, Price emphasized a “physical definition of selection” as the basis for the
general model of selection and focused on the “act or process” of selection rather than
“preference, optimization calculation, or decision making that may have given rise to it”
(p. 390). According to Price’s general model, any forms of selection (1) operate on a
population or a set of individuals, each of which is characterized by some value of trait 𝑧 𝑖
and (2) produce a corresponding set whose overall characteristic differs from that of the
original set, that is, 𝑧 ̅ ≠ 𝑧 ̅
′
. The difference in overall characteristic between the two sets is
expected if and only if individuals with different traits are selected at different rates, that
is, 𝑤 𝑖 ≠ 𝑤 𝑗 . Therefore, the variance in selection rate among individuals with different
traits 𝑣𝑎𝑟 (𝑤 𝑖 ) indicates the intensity of selection, and the covariance between selection
rate and trait value 𝑐𝑜𝑣 (𝑧 𝑖 ,𝑤 𝑖 ) indicates the orientation of selection, as described in the
Price equation.
To demonstrate, consider shopping for apples at a store as an example, which
does not involve any evolutionary processes of biological meanings. Suppose that there
are 12 apples in the store, which can be grouped by their size 𝑧 . More specifically, four of
them are relatively small (say, 2 inches in diameter), and the other eight are large (3
inches in diameter). Further suppose that shopper X bought one small and two large
apples, whereas shopper Y bought two small and two large apples from the store. At the
first glance, shopper X seemed to prefer large apples over small ones, because he chose
large apples more than small ones. However, the selection made by shopper X was
actually unbiased or random in terms of the size of apples, because he chose each apple
with an equal probability regardless of their size. Shopper X picked exactly 25% of small
apples (one out of four) and 25% of large apples (two out of eight). There was neither
43
difference nor variance in selection rate between the two groups, 𝑤 𝑧 =2
= 𝑤 𝑧 =3
=
.25; 𝑣𝑎𝑟 (𝑤 𝑖 ) = 0, and therefore, no change in the overall characteristics, ∆𝑧 ̅ = 𝑧 ̅
𝑋 ′
− 𝑧 ̅ =
2.66− 2.66 = 0. On the other hand, shopper Y seemed to be indifferent between large
and small apples, because she chose the same number of large and small apples. But the
selection made by shopper Y resulted in a substantial change in the overall characteristic.
The average size of all the apples before selection was 2.66, while that of the selected
apples was 2.50 (i.e., ∆𝑧 ̅ = 𝑧 ̅
𝑌 ′
− 𝑧 ̅ = 2.50− 2.66 = −0.16). This is because shopper Y
chose small apples more likely (𝑤 𝑧 =2
= .50) than large apples (𝑤 𝑧 =2
= .25), which is
confirmed by the negative covariance between selection rate and size of apples
𝑐𝑜𝑣 (𝑧 𝑖 ,𝑤 𝑖 ) = −.50. Figure 2.2 illustrates this example.
Price’s general model of selection allows us to form the inference about the
direction and intensity of selection bias by comparing two corresponding sets of objects:
one is the set of the selected objects; and the other is the original set from which the
selected objects are selected. In the above example, the selection biases of the two
shoppers cannot be properly evaluated by looking at the size distribution of the selected
apples without comparing it with that of the apples in the original set. Although shopper
X chose more large apples than small ones, this does not necessarily imply that the
shopper preferred large apples over small ones. This is because large apples were much
more common in the original set. If every apples in the original set was chosen uniformly
at random, that is, with an equal probability, the set of the apples selected by shopper X
will be most likely obtained. Therefore, shopper X’s selection was unbiased or random.
44
Figure 2.2. The application of the general model of selection to shopping for apples
(adapted from Price 1995 p. 390).
Note: Shopper X chose both large and small apples at an equal rate (𝑤 𝑧 =2
= 𝑤 𝑧 =3
= .25),
resulting in no substantial change in the average size of apples (∆𝑧 ̅ = 𝑧 ̅
𝑋 ′
− 𝑧 ̅ = 0). On the
other hand, shopper Y chose small apples more likely than large apples (𝑤 𝑧 =2
= .50 >
.25 = 𝑤 𝑧 =3
), decreasing the average size of apples (∆𝑧 ̅ = 𝑧 ̅
𝑌 ′
− 𝑧 ̅ = −.16).
2.2.4. Moran Process and Neutral Drift
The Moran process (Moran, 1962) represents the simplest possible stochastic
model to study the three evolutionary mechanisms—replication, mutation, and
selection—in a finite population (Figure 2.3). Under the assumption that a population of
interest has reached the limited carrying capacity of its environment, the population
grows only when existing individuals are eliminated. In each step, one individual is
chosen for elimination at random (the white circle), and another individual is chosen for
45
replication (with replacement) to fill up the empty spot. In that case, some individuals
(e.g., the grey squares) may be fitter and more likely to reproduce than others (e.g., the
black circles). As a result, the relative frequency of the fitter population becomes larger
than that of the other (the upper box in the third column). However, replication is not
always perfect but often produces new variants (i.e., the triangle), increasing the trait
diversity of the population.
Figure 2.3. The Moran process in a finite population.
As a stochastic model, the Moran process capture an important feature of
evolutionary dynamics in finite populations, namely, neutral drift. To illustrate, suppose
a finite population that consists of 10 individuals, each of which is characterized by either
of two distinct traits (e.g., square and circle), and in which only replication occurs in the
absence of mutation and selection. In this population, both the variety of the population
and the relative frequencies of subpopulations are supposed to remain the same over time.
46
However, the relative frequencies of subpopulations can change because of random errors
in sampling. In this example, there are five squares and five circles at time t. Thus, a
square could be chosen either for elimination or replication with a probability of .50, and
a circle could be chosen with the same probability. There are four possibilities of what
could happen between times t and t + 1: (a) a square could be chosen for elimination and
replication, (b) a square could be chosen for elimination and a circle for replication, (c) a
circle could be chosen for elimination and replication, or (d) a circle could be chosen for
elimination and a square for replication. Each of these four events can occur with an
equal probability of .25 (Figure 2.4).
Figure 2.4. The likelihood that the relative frequencies of subpopulations change in the
absence of mutation and selection.
Notice that the relative frequencies of the two subpopulations remain the same in
the cases of (a) and (d), whereas they change in the cases of (b) and (c) despite the
absence of mutation and selection. The total probability of the latter is .50 (= .25 + .25).
In general, the frequency distribution of subpopulations could change over generations
purely by chance. Such a change is called “neutral drift” (Nowak, 2006) and should not
47
be considered as the consequences of selection. The effects of neutral drifts are not trivial
when a finite number of individuals are selected for replication or elimination and are
maximal when only one individual is selected by the law of large numbers.
2.3. The Evolutionary View of Human Rationality
Because of its theoretical utility the evolutionary theory has been widely adopted
as a general framework for the dynamic change of complex systems across disciplines.
However, there has been skepticism about its applicability to social and cultural systems
in human societies (Boyd & Richerson, 2005). This is in large part because of the beliefs
in human rationality and agency (Gigerenzer, 2008). Unlike animals or any other subjects
of evolution, people would not blindly repeat what their predecessors did but instead
purposively try something new on the basis of their own rational calculation. Therefore,
neither exact replication nor random mutation is expected in the evolution of social and
cultural systems. All human behavior is as the results of exercising their free will, and
therefore, individuals are the key player in the course of the evolution of human societies.
The belief in human rationality however has been challenged by many philosophers, such
as Immanuel Kant (1781/2007) and Friedrich Nietzsche (1886/1997), and still remains as
one of the most controversial issues in social science and philosophy (Coleman & Fararo,
1992).
Interestingly, the debate over human rationality entered into a new phase, when
John Maynard Smith introduced evolutionary game theory in 1973. As its name itself
implies, evolutionary game theory is the evolutionary version of game theory. Game
theory is the study of conflict and cooperation among strategic decision-makers and has
been developed in theoretical economics and behavioral science (Fudenberg & Tirole,
48
1991; von Neumann & Morgenstern, 1947). Games represent various social situations in
which individuals’ payoffs are jointly determined by their own decisions and the
decisions made by their competitors (e.g., negotiation and bargaining). In classical game
theory (von Neumann & Morgenstern, 1947), individuals are assumed to be rational, self-
interested, and able to make decisions that maximize their own payoffs (e.g., whether to
cooperate or to exploit others). A primary goal of classical game theory is to make
analytical predictions about what action will be taken collectively in games under the
assumption of players being rational. For example, Nash equilibrium is one of the best
known generalized predictions of classical games, defined as a state in which no player
has anything more to gain by changing only their own strategy (Nash, 1951). However,
game theory has been challenged because in many real-world situations human players
do not actually behave in the way that the theory predicts (Camerer, 2003). Some
scholars attribute the poor predictability of game theory to its unrealistic assumption
about human rationality (Colman, 2003; Luce & Raiffa, 1957). For example, Gigerenzer
and Selten (2001) argue that it is practically impossible to collect and process all the
necessary information to draw a rational decision, and further, it is unnecessary, even if
possible, following Herbert Simon’s (1959) notion of bounded rationality.
Maynard Smith’s (1973; 1982) evolutionary game theory provides a new
perspective to understanding the dynamics and complexity embedded in game-like
situations. As an evolutionary biologist, Maynard Smith was interested in the conflicts
and competition among animals. In particular, he was puzzled by the counter-intuitive
fact that animals often cooperate with others who are potential competitors in the struggle
for survival and that those cooperative animals sometimes out-survive the others over
49
time (Hamilton, 1963; Trivers, 1971). To understand cooperative and even self-
sacrificing behavior of animals, he adopted the framework of classical game theory but
made three important modifications. First, evolutionary game theory does not require
individuals to act rationally simply because animal players cannot do so. Instead,
individuals always play strategies or behavioral phenotypes inherited from their parents.
In this case, the strategies are seen as heritable traits, and individuals can be divided into
subpopulations by the strategies they play. Second, payoff, which is usually interpreted as
material reward in classical game theory, is understood as growth rate (or Darwinian
fitness), and success in a game is translated into reproductive success. Individuals with
strategies that do well reproduce faster than others, and thus, more individuals will play
those strategies in the next generation. Final and most important, games are played
repeatedly, in particular, over generations. Notice that individuals do not consciously
choose one strategy over the others in evolutionary games. Instead, the natural
environment surrounding them selects strategies based on their fitness. This is
straightforward natural selection.
An evolving population of individuals with different strategies will end up with
either the dominance of a single strategy (i.e., survival of the fittest) or the coexistence of
two or more strategies (i.e., survival of many). To determine the final state of a
population, Maynard Smith (1982) introduced a new concept, the evolutionary stable
strategy (ESS) and defined it as “a strategy such that, if all the members of a population
adopt it, then no mutant strategy could invade the population under the influence of
natural selection” (p. 10, see also Duersch, Oechssler, & Schipper, 2012). If one of the
available strategies is an ESS, it is likely that the strategy will take over the entire
50
population, meaning that only the individuals who play that strategy will survive. If none
of them is an ESS, multiple strategies will coexist in the population, which is more
common in real environments. This particularly implies that cooperative and altruistic
behaviors can survive even in highly competitive environments.
The evolutionary game-theoretic account for cooperation and altruism in animals
provides important insights into understanding the emergence and stabilization of certain
actions and behaviors at the population level in human societies, such as social norms,
conventions, and cultures (Boyd & Richerson, 2005). Most real-world social and
collective phenomena involve much more players and strategies than the canonical forms
of games do (e.g., two-person, two-strategy prisoner’s dilemmas), and the complexity and
uncertainty inherent in real-world problems well exceed the abilities of “rational” players
to deal with. In fact, many real-world problems cannot be solved by analytical methods
used in classical game theory. For instance, it is well known that no analytical methods
can find an explicit solution for the situation in which two strangers pass on a narrow
hallway in opposing directions without blocking each other (i.e., coordination games,
Coleman, 1973; Cooper, 1999; D. Lewis, 1969). For this problem, evolutionary game
theorists have proposed to replace rational decision making with a set of rather simple
behavioral rules, such as repeating the successful strategies in the past, learning by
imitating successful others, and following the majority (Gigerenzer, 2008; Harms &
Skyrms, 2008). Many theoretical and empirical studies have shown that those simple
rules successfully reproduce a variety of social and collective phenomena observed in
human societies. A small sampling of topics which have been analyzed from the
evolutionary perspective include altruism (Fletcher & Zwick, 2007; Gintis, Bowles,
51
Boyd, & Fehr, 2003), behavior in public goods games (Doebeli & Hauert, 2005; Hauert
& Imhof, 2012), empathy (Fishman, 2006; Page & Nowak, 2002), moral behavior
(Harms & Skyrms, 2008; Skyrms, 2004), private property (Gintis, 2007), language
(Nowak, Plotkin, & Krakauer, 1999; Skyrms, 2010), and social norms (Axelrod, 1986;
Ostrom, 2000).
Note that Maynard Smith’s evolutionary game theory and many of its applications
do not address the questions about whether, to what extent, or in what way human
rationality plays out in the evolution of social and cultural systems. But instead, they
show that a wide range of social and collective phenomena can be explained without the
rationality assumption when the concept of human rationality is replaced by that of
evolutionary stability. Of course, this does not necessarily mean that human players do
not (or cannot) act rationally to optimize their collective outcomes. What Maynard Smith
suggested is an alternative shortcut to understanding social and collective phenomena
without going through all the considerations of purposiveness or intentionality of
individual players. He made this point clearer by using an analogy with physical theory:
When calculating the path of a ray of light between two points, A and B, after
reflection or refraction, it is sometimes convenient to make use of the fact that the
light follows that path which minimizes the time taken to reach B. It is a simple
consequence of the laws of physics that this should be so; no one supposes that
the ray of light setting out from A calculates the quickest route to B. Similarly, it
can be a simple consequence of the laws of population genetics that, at
equilibrium, certain quantities [of traits] are maximized. If so, it is simplest to find
the evolutionary stable equilibrium by performing the maximization. Nothing is
52
implied about intention, and nothing is asserted about whether or not the
equilibrium state will favor species survival (1982, p. 5).
2.4. The Application of Evolutionary Mechanisms to Network Dynamics
Thus far, the current chapter have discussed the major concepts of evolutionary
theory, including the three fundamental mechanisms of evolutionary dynamics—
replication, mutation, and selection. The next two chapters propose an evolutionary
model of scientific collaboration networks based on the three mechanisms. Successful
applications of evolutionary theory require adequate translation of the three mechanisms
in the context of scientific collaboration networks. Thus, the question to be addressed to
develop a model of network evolution is how to interpret those mechanisms in the
context of network dynamics. This section discusses the applicability of evolutionary
theory to network dynamics by revisiting previous models of network growth from the
evolutionary perspective. Three models to be reviewed are Barabási and Albert’s (1999)
scale-free network model, Watts and Strogatz’s (1998) small-world network model, and
Kleinberg et al.’s (1999) vertex copying model.
2.4.1 The Scale-free Network Model
Barabási and Albert’s (1999) scale-free network model is composed of two
growth processes. One is the preferential attachment process by which edges are formed
between existing nodes and newly arriving nodes. The other is the continuous addition of
new nodes to an existing network. When these two processes occur simultaneously, the
degree distribution of the network follows a power-law distribution.
The preferential attachment process and the continuous addition of nodes can be
interpreted as replication and mutation processes, respectively. To demonstrate, consider
53
all the edges of a network at a given moment as a population of edges. In the scale-free
model every edge is formed between an existing node and a new node. Therefore, one of
the two nodes that an edge is incident upon is always older than the other. Then,
individual edges can be characterized by their older nodes, and further, the population of
edges can be divided into subpopulations. When subpopulations are defined in this way,
there will be as many subpopulations as nodes in the network.
Now suppose that every subpopulation of edges grows at an equal rate of w by
simple replication under an unlimited supply of resources without influencing or being
influenced by the growth of every other. Then, the growth of subpopulation i can be
described as:
𝑑 𝑥 𝑖 𝑑𝑡 = 𝑤 𝑥 𝑖 .
(2.9)
where 𝑥 𝑖 is the size of the ith subpopulation and w is the common reproduction rate (see
Section 2.2.1). Notice that 𝑥 𝑖 is the total number of edges incident upon the ith node and
therefore is the degree of node i by definition, which can be denoted by 𝑘 𝑖 . By
substituting 𝑥 𝑖 with 𝑘 𝑖 , the above equation can be rewritten as:
𝑑 𝑘 𝑖 𝑑𝑡 = 𝑤 𝑘 𝑖 .
Further, if the reproduction rate w is equal to 1/∑𝑘 𝑗 𝑗 , meaning that one edge is formed
in the network at a time, the above equation becomes exactly identical to the equation for
the preferential attachment process 𝑑 𝑘 𝑖 /𝑑𝑡 = 𝑘 𝑖 ∑𝑘 𝑗 𝑗 (Equation 1.1).
Of course, the replication of edges would not be always prefect. The failure in
replication will give rise to new types of edges that are incident upon brand-new nodes.
54
That is, the mutation of edges can be interpreted as the introduction of new nodes, which
is the second mechanism of the scale-free network model.
It is worth noting that the scale-free network model does not involve any
substantial selection mechanism, although the preferential attachment process is often
misinterpreted to imply selection bias in the formation of edges. Notice that every
subpopulation of edges grows at an equal rate of w regardless of their characteristics
(Equation 2.9), and the variance in growth rate among the subpopulations is therefore
zero, 𝑣𝑎𝑟 (𝑤 𝑖 ) = 0, which implies the absence of selection bias, according to Fisher’s
theorem and the Price equation (Equation 2.8).
This becomes clearer when we consider other power-law distributions that arise
by preferential attachment processes. For example, the distribution of family names is
known to arise by a preferential attachment process and to follow a power-law (Reed &
Hughes, 2003). Family names are duplicated and passed on from parents (males only in
many cultures) to their children. The number of individuals with a family name in a
population increases whenever any of those individuals come to have a child. Suppose
that an individual is sampled to have a child uniformly at random as in the Moran
process. The probability that the sampled individual (and his child) has a particular
family name is proportional to the number of individuals with that name in the
population. Hence, the frequency of a family name increases proportionally to the
frequency of the name at a given moment. When such a process is repeated over
generations, frequent family names will becomes more frequent, and the distribution of
55
family names approaches a power-law. Note that this happens because the selection is
made cumulatively but not because it is biased.
8
2.4.2 The Small-World Network Model
Watts and Strogatz’s (1998) proposed the random rewiring process as the
generative mechanism of small-world networks characterized by short average path
lengths and high levels of transitivity. The model parameter p, denoting the proportion of
rewired edges, controls the interpolation between a regular network and a random
network. When p = 0 no edges are rewired, retaining the original regular network. The
regular network is as transitive as real-world networks, but its average path length is
longer than those of real-world networks. When p = 1, all edges are rewired to random
positions, resulting in a random network. The average path length of the random network
is close to those of real-world networks, but the random network is not as transitive as
real-world networks. For intermediate values of p, small-world networks can be
generated, which well reserve both properties of real-world networks—the high levels of
transitivity and short average path lengths.
Now consider individual nodes and the patterns of their linkages with their
immediate neighbors (i.e., ego-centric networks) in the three types of networks—the
regular, random, and small-world networks. The regular network can be seen as perfectly
ordered, because all nodes share a common linkage pattern, as if one’s pattern were an
identical copy of another’s. The regular network can grow without any substantial
8
The interested reader who wishes to pursue this matter is referred to the very first formulation of
preferential attachment processes by Udny Yule (1925), who used it to explain the emergence of the power-
law distribution of the number of species per genus of flowering plants only by replication and mutation
processes.
56
structural change, as nodes with the identical linkage pattern are newly added to the
network, that is, through exact replication. On the other hand, the random network is
disordered or chaotic, because every single node, in principle, has a unique linkage
pattern. In the small-world network characterized by an intermediate value of p, a
majority of nodes share a common linkage pattern as the results of replication, whereas a
few others have variant patterns as the results of the random rewiring or the failures of
exact replication. The dominance of the common linkage pattern maintains the high level
of transitivity of the regular network, whereas the variation in the linkage patterns
reduces the average path length of the network by creating shortcuts. In this sense, the
small-world network can be seen as a byproduct of replication of linkage patterns with
non-trivial mutations. Further, it can be seen as to be on or near to “the boundary between
order and chaos, a state which optimizes the complexity of tasks the systems can perform
and simultaneously optimizes stability” (Kauffman, 1993, p. 173).
However, the small-world network model does not contain any substantial
selection mechanism, either. When all the nodes in a network are considered as a
population, the node population can be divided into subpopulations by their linkage
patterns. In the regular network, there is one and only one subpopulation, because every
node has the same linkage pattern. On the other hand, there could be as many
subpopulations as the nodes in the random network, because each has a unique linkage
pattern. In the small-world network, a majority of nodes have a common linkage pattern,
while a few have unique linkage patterns. Because existing edges can be replaced
between any pairs of nodes by the random rewiring, any linkage patterns are equally
probable in the random and small-world networks, and thus, all possible subpopulations
57
are equally likely to be realized (i.e., 𝑤 𝑖 = 𝑤 𝑗 for all 𝑖 ≠ 𝑗 ). This implies the absence of
selection bias.
2.4.3 The Vertex Copying Model
Recent interest in scale-free networks has apparently started with work by
Barabási and Albert (1999), but its generative mechanism, which they called “preferential
attachment,” was essentially the same as that proposed by Derek John de Solla Price
(1965) to explain the occurrence of power-law degree distributions in the networks of
citations between scientific papers, which he called “cumulative advantage.” According
to Derek Price, a researcher perusing the literature in a given academic field would
encounter citations to frequently cited papers more often than citations to less cited ones
and hence would be more likely to cite those frequently cited papers. Another way of
saying this is that researchers are “copying” citations from the bibliographies of papers
they read.
9
Kleinberg et al. (1999; see also Kumar et al., 2000) have proposed the vertex
copying model that takes this idea and applied it to the growth of the Web, the network of
hyperlinks among webpages.
The copying process of the model describes the behavior of webpage creators
building hyperlinks from their pages to existing pages in a similar way of scientific
researchers “copying” citations from the bibliographies of papers they read. Potential
webpage creators would encounter webpages that contain hyperlinks to other webpages
relevant to the topics of interest to them and hence would include many of those links in
9
The word “copying” here is used figuratively, but in fact there is evidence suggesting that researchers
really do just copy citations form other papers, possibly without even looking at the cited papers. For
instance, it was often observed that many different authors used the same wrong page number in citing a
particular paper (Simkin & Roychowdhury, 2006).
58
their webpages. A more precise description of the copying process is as follows. (1) A
network grows as new nodes arrive and form edges to existing nodes in the network. (2)
Each new node i chooses an existing node j uniformly at random and copies j’s linkage
pattern. (3) New nodes, however, often do not make the exact copies of the linkage
patterns of chosen nodes, producing variant linkage patterns. Figure 2.5 illustrates the
copying process.
Figure 2.5. The vertex copying process, where i is a new node (the grey node), and j is an
existing node (the blue node) chosen at random for the replication of its linkage pattern
(the blue solid lines) with mutation (the red dashed line).
Notice that the second step of the copying process is, in effect, identical to the
preferential attachment process: the probability that an existing node j is connected to
another existing node k is proportional to k’s degree. When a new node i arrives and
chooses an existing node j uniformly at random to copy its linkage pattern, the
probability that node i is connected to k is equal to the probability that node j is connected
to node k, which is proportional to k’s degree. That is, the probability that node k receives
59
a new edge from node i is proportional to its degree at a given moment as in the
preferential attachment process. Hence, the repetition of the copying process eventually
leads to the occurrence of a power-law degree distribution in the network.
When nodes of a network and their linkage patterns are considered as individuals
of an evolving population and a particular trait that characterizes the individuals,
respectively, the vertex copying process can be understood as a special kind of the Moran
process. At each point in time, a node is chosen for the replication of its linkage pattern,
and mutation occurs when the replication fails and produces new linkage patterns. One
difference is that individuals are chosen for replication at different rates according to their
fitness in the Moran process, whereas nodes are chosen for replication uniformly at
random in the vertex copying model. In other words, the vertex copying process does not
involve any substantial selection mechanisms, either.
To summarize, the underlying mechanisms of the three network growth models
can be interpreted as replication and mutation mechanisms in the context of network
evolution. Specifically, the preferential attachment process can be understood as the
growth of edges by replication, the random rewiring process as the increase in the
variation of linkage patterns of ego-centric networks by mutation, and the vertex copying
process as the replication of existing nodes’ linkage patterns with mutation. However, the
models do not contain any theoretical components that correspond to the selection
mechanism of the evolutionary dynamics. All forms of selection in those models are
made unbiasedly or uniformly at random.
However, the selection mechanism is critical to understand the growth dynamics
of scientific collaboration networks. In scientific collaboration networks, nodes are not
60
just any authors who are chosen uniformly at random. But instead, they are the authors
who have produced acceptable papers. Similarly, edges are not formed between any pairs
of authors chosen at random. They can be formed only between those who have worked
together to produce acceptable papers. In that sense, scientific collaboration networks are
not just growing but instead evolving over time, which may not fully explained by
previous growth models. From this perspective, the following chapter develops a
selection model of collaboration networks as a plausible explanation for their structural
change.
61
CHAPTER 3: THE SELECTION MECHAINSM OF SCIENTIFIC
COLLABORATION NETWORKS
Scientific collaboration networks grow over time as new authors enter and new
collaboration links are formed, when networks are constructed by all the papers ever
published in journals. The addition of new authors and collaboration links leads to the
structural change of collaboration networks, although the change may or may not be
substantial. For example, the numbers of nodes and edges of a collaboration network
increase exactly by the numbers of new authors and new collaboration links, respectively.
The network becomes more interconnected than before when new collaboration links are
formed between authors from different clusters. What if a different set of authors and
their collaboration links were added to the existing network? Obviously, the structure of
the collaboration network would have changed in a different way. For example, the
collaboration network would have been smaller in size than it is if single-authored papers
were selected more than multiple-authored papers. A new isolated cluster could have
been formed, and thus, the collaboration network would have been less interconnected
than it is, if a team of authors who have no collaborative connections to anyone else
entered as new nodes. The structural properties of collaboration networks vary depending
on which set of authors and collaboration links are added. Further, a collaboration
network observed at a given moment is one of many possible networks into which it
would have changed.
Although authors could submit any papers and collaborate with anyone as they
wish, not all authors or their collaborations will appear in collaboration networks. Only
those whose papers have been accepted for publication can enter the collaboration
62
networks and form connections only to their coauthors. This implies that there is some
mechanism that (1) selects a particular set of authors and their collaboration links out of
many possible sets and thereby (2) determines the course of the structural change of
collaboration networks by adding the selected set of authors and collaboration links to the
networks. Then, the question to address becomes “by what selection mechanism would a
collaboration network come to have its observed structural properties?” To address this
question, this chapter proposes a theoretical model of the selection mechanism that
determines the direction of the structural changes of collaboration networks.
3.1. A Snapshot of the Evolution of Scientific Collaboration Networks
Before proceeding, it would be helpful to look at the structural change of an
empirical collaboration network. The following example describes the structural change
of the collaboration network of the Journal of Communication between 2009 and 2010.
In 2010 a total of 41 papers was accepted for publication. Of them, 21 papers were
written by single authors (51.2%), 12 papers by two authors (29.3%), 6 papers by three
authors (14.6%), and two papers by four authors (4.9%). A total of 68 distinctive authors
wrote the 41 papers. Of them, 57 were experienced authors who had published one or
more papers before 2010 (i.e., repeated authors: 82.8%), whereas the other 11 were those
who have no previous publication experience before 2010 (i.e., new authors: 17.2%). A
total of 42 collaborative connections were found among the 68 authors, 30 of which were
preexisting and repeated collaboration links (i.e., repeated collaborations: 72.4%), and the
other 12 were newly formed in 2010 (i.e., new collaborations: 27.6%).
Figure 3.1 visualizes the structural change of the collaboration network of the
Journal of Communication between 2009 and 2010. Note that the networks in the figure
63
include only the largest connected components for illustrative purpose. The colored nodes
and links on the right panel represent the authors and their collaboration links of the
papers published in the journal in 2010. More specifically, the blue nodes are the repeated
authors, while the blue links are the repeated collaborations. On the other hand, the green
nodes and links are the new authors and collaborations in 2010, respectively. The size of
nodes indicates the cumulative number of publications of the corresponding authors in
2010. The addition of a set of new nodes and links leads to the structural change of
collaboration networks. Most immediately, the numbers of nodes and edges of the
collaboration network increased. Accordingly, a variety of structural properties of the
network changed.
(a) The collaboration network in 2009 (b) The collaboration network in 2010
Figure 3.1. A snapshot of the structural change of the collaboration network of the
Journal of Communication between 2009 and 2010 (the largest connected components
only).
According to the International Communication Association, the publisher of the
journal, the acceptance rate of the journal was roughly 11% (International
64
Communication Association, 2014). That is, about 370 manuscripts were submitted to the
journal but most of them (89%) were rejected in 2010. This means that there were a large
number of other possible sets of new nodes and links that might have been added, each of
which would have resulted in different structural changes. Therefore, the collaboration
network observed in 2010 (Figure 3.1b) is just one of many possible networks into which
it would have changed if different nodes and links were added. Clearly, the publication
decisions made by the journal determined which nodes and links to add. In this sense
journals play a significant role as the agent of selection in the evolution of scientific
collaboration networks. For example, if a journal publishes a large number of papers
every year, the size of the collaboration network will grow rapidly. If a journal prefers
papers by experienced authors, repeated authors and repeated collaboration links will be
common in the network, and the network will become denser without new nodes being
added.
The selection mechanism operated by academic journals is an important building
block for an evolutionary model of scientific collaboration networks, which is discussed
in this chapter. The selection mechanism proposed here is based on the stochastic process
by which knowledge elements retrieved from individual authors are combined and
generate knowledge products (i.e., journal papers in this case), suggested for the detection
of scientific communities by Gilbert (1997) and applied as the team assembly mechanism
in scientific collaborations by Guimerà, Uzzi, Spiro, and Amaral (2005). The stochastic
process is further modified to explore the impacts of the selection criteria of journals on
the structural properties of collaboration networks.
65
3.2. Defining Papers and the Selection Criteria of Journals
Most academic journals adopt peer-review processes to make publication
decisions on submitted manuscripts (Kronick, 1990). Therefore, peer-review processes,
more precisely, the decisions made by journal editors and reviewers are an important part
in the selection mechanism of the evolution of collaboration networks. However, the
selection mechanism should not be reduced to the peer-review processes at least for two
reasons. First, it is not easy to empirically measure the peer-review processes, in
particular, to identify their selection criteria. A number of previous studies have
attempted to specify the selection criteria adopted by journals by analyzing empirical
data. Examples include content-analysis to compare between accepted and rejected
manuscripts (Peters & Ceci, 1982), discourse analysis of correspondences between
editors, reviewers, and authors (Berkenkotter & Huckin, 1995), and in-depth interviews
with editors and reviewers (Kuper, Reeves, & Levinson, 2008). However, previous
studies failed to find any concrete or consistent selection criteria even within single
journals (for a critical review see Rowland, 2002).
Second and more important, the peer-review processes do not fully represent the
selection of papers, even if the selection criteria of editors and reviewers are fully
identified. Note that publication decisions are made only among the papers that have been
submitted. If authors decide not to submit their papers to a particular journal, the papers
will never be published regardless of whatever criteria the journal adopts. If that is the
case, the selection criteria of journal editors and reviewers cannot fully account for why
some papers were not selected for publication. Note that the publication decisions of
papers are made jointly in part by journal editors and reviewers but also by authors. It is
66
not uncommon that some manuscripts are not submitted even when they are ready for
review and qualified enough for publication. For example, Thomas Bayes (1702-1761)
did not submit his paper entitled “An Essay towards Solving a Problem in the Doctrine of
Chances” (1763). The manuscript was discovered and submitted by his friend Richard
Price two years after his death and eventually became one of the most important works in
probability theory. However, there is no way to find out the reason why Bayes did not
submit the paper or how many more papers he wrote but did not submit.
What is necessary for building a selection model is the “act or process” of
selection rather than the “preference, optimization calculation, or decision making that
may have given rise to it” (G. Price, 1995, p. 390). For understanding the evolutionary
processes of collaboration networks, it would matter little to know what kinds of papers
journals prefer or why authors decide to submit their papers or not. From this perspective,
the model proposed in this chapter is rather general and abstract so that it does not rely
upon any context-specific assumption or any reference to objective reality. Nevertheless,
it allows us to draw reasonable conclusions about the consequences of the selection
mechanism on the structure of collaboration networks, namely, the “analysis without
measurement” (Katzner, 1983).
To begin, consider a way to define academic papers. An academic paper is
characterized by a unique combination of various elements, such as its theoretical
foundations, analytical methods, data used, references, writing style, and so forth. When
each of the elements can be denoted by a distinctive value (either numeric or non-
numeric), the paper can be represented by a sequence of those values, which Gibert
(1997) called kenes of knowledge products, a neologism intentionally similar to “genes.”
67
In other words, a paper can be defined by an h-tuple vector, 𝐳 = [𝑧 1
,𝑧 2
,𝑧 3
,…,𝑧 ℎ
], each
of whose elements represents the knowledge element of the paper in the corresponding
dimension. Hence, a paper can be graphically presented as a point in an h-dimensional
space, whose coordinate is [𝑧 1
,𝑧 2
,𝑧 3
,…,𝑧 ℎ
]. If two papers’ sequences are identical, and
thus, if they are presented as the same point, one is an exact copy of the other. In reality,
however, no two papers would exist whose sequences are perfectly identical due to the
common principle of anti-plagiarism in the academic community. Therefore, every
publishable paper is assumed to be denoted by a unique sequence and be presented as a
unique point in an h-dimensional element space.
Next, consider the process in which a paper, or more precisely, its sequence of
knowledge elements is produced either by single authors or through the collaboration of
multiple authors. Further, assume that the knowledge elements of a paper are originally
“stored” in authors of that paper and retrieved and combined during collaboration. A
paper could be in principle represented by a sequence of any length. For purely
demonstrative purposes, however, let us limit the allowable length of sequences to 2,
(i.e., 𝐳 = [𝑋 ,𝑌 ]) so that the process can be illustrated on a two-dimensional plane (Figure
3.2). For example, a value of x may represent the primary theoretical framework that a
paper adopts, while a value of y may represent the analytical method used in the paper.
The paper can be characterized by the combination of its theoretical framework and
analytical method, denoted by a pair of values [𝑥 ,𝑦 ], and presented as a point on a two-
dimensional plane. When x and y are real numbers, there are infinitely many possible
values in each dimension and thus an infinitely number of possible combinations of
knowledge elements, even though the allowable length of sequences is limited to 2.
68
Suppose that author i has a finite number of knowledge elements, say three, on the
first knowledge dimension 𝐱 𝑖 = [𝑥 1
,𝑥 2
,𝑥 3
]
𝑇 and three knowledge elements on the other
knowledge dimension 𝐲 𝑖 = [𝑦 1
,𝑦 2
,𝑦 3
]
𝑇 . This means that the author is knowledgeable
about three different theories 𝑥 1
, 𝑥 2
, and 𝑥 3
and competent with three different analytical
methods 𝑦 1
, 𝑦 2
, and 𝑦 3
. On the other hand, author j has three different knowledge
elements on the first knowledge dimension 𝐱 𝑗 = [𝑥 4
,𝑥 5
,𝑥 6
]
𝑇 and another three
knowledge elements on the other knowledge dimension 𝐲 𝑗 = [𝑦 4
,𝑦 5
,𝑦 6
]
𝑇 . Each author
alone can generate nine unique combinations (= 3 × 3) by using his or her own
knowledge elements, each of which represents a unique paper under the assumption that
the number of knowledge elements that an individual author possess does not change
over time. The nine papers that author i can generate are presented as the blue dots, while
another nine papers that author j can generate are presented as the red dots in Figure 3.2.
If the two authors collaborate, they can generate 18 additional unique papers (the green
dots in Figure 3.2) because they can produce the combinations of knowledge elements by
exchanging their unique knowledge elements which they could not have produced
individually.
A journal’s selection criteria can be defined in a similar manner. But they will be
represented by a two-dimensional object having an area rather than a single point. To
define a journal’s selection criteria, consider all the papers that are potentially acceptable,
including those that have been already accepted by the journal. Each of those papers can
be denoted by a unique sequence of knowledge elements and presented as a point on the
plane as before. As long as the journal is assumed to have consistent selection criteria, the
points of acceptable papers are expected to be positioned close to one another forming a
69
cluster rather than being randomly scattered all over the plane (Figure 3.3). Such a cluster
can be called the “sweet spot” of the journal, and its boundary determines the range of
acceptable combinations of knowledge elements.
Figure 3.2. The possible combinations of knowledge elements by single authors (the red
and blue dots) and through collaboration (the green dots).
Figure 3.3. A journal’s sweet spot defined as the collection of all acceptable papers.
70
By the definition of sweet spot, the fact that a newly written paper is located
within the sweet spot of a journal implies that the paper is one of the acceptable papers
and contains all the knowledge elements required to be published in the journal. Such a
paper is more likely than the others to be submitted to and accepted by the journal. This
in turn implies that there is a nontrivial difference in the rate of selection for publication
between papers within the sweet spot and the others, and thus, the presence of selection
pressures following Price’s (1995) model. The characteristic of papers associated with the
different selection rate is the location on the two-dimensional plane, and its coordinates
denote specific combinations of knowledge elements. In short, papers will be submitted
and accepted at different rates depending on the knowledge elements that compose the
papers: i.e., 𝑣𝑎𝑟 (𝑤 [𝑥 ,𝑦 ]
) > 0.
3.3. The Impacts of Selection Criteria on Network Structure
3.3.1. The Impacts of Selection Criteria at the Dyadic Level
Now consider the impacts of the selection criteria of journals—that is, the
location, size, and shape of their sweet spots—on the structure of the collaboration
network of its authors. First, consider three hypothetical journals and the impacts of their
selection criteria on the dyadic level. In the previous two-author example, suppose
journal A whose sweet spot is on the bottom left of the plane (the purple ellipse in Figure
3.4). Of the 36 possible papers only three papers fall into the sweet spot, and author i
alone can produce those three papers. Author i’s name is expected to appear in the journal
up to three times. However, it does not necessarily mean that all the papers written by
author i will be accepted by the journal. Six of the nine papers the author can produce
will not be accepted because there are out of the sweet spot. When his or her papers are
71
submitted and accepted for publication, author i will be represented by an isolated node
in the collaboration network, because the three papers involve no collaboration (Figure
3.5a). None of the papers that author j can produce are located in the sweet spot of
journal A. Thus, author j would not submit any papers to A, or the author’s paper will be
rejected, even if the author submits one. On the other hand, another journal B, whose
sweet spot is on the bottom right of the plane (the yellow ellipse in Figure 3.4) can result
in a different network structure. The two authors can produce acceptable papers only if
they collaborate. As a result, the two authors’ names will always appear together in the
journal, and they will be represented by a connected dyad in the collaboration network
(Figure 3.5b). In that case, their collaboration will be repeated up to four times. In the
case of journal C, whose sweet spot lies around the center of the plane (the grey ellipse in
Figure 3.4), the two authors can produce acceptable papers individually, but their
collaboration will not help to produce any acceptable papers. Therefore, it is expected
that both authors’ names will appear in the journal, but they will be represented by a
disconnected dyad (Figure 3.5c).
72
Figure 3.4. Selection of papers (dots) for publications by journals’ selection criteria
(ellipses).
(a) Journal A
(b) Journal B
(c) Journal C
Figure 3.5. The expected forms of dyads in three hypothetical journals.
This simple example demonstrates two important points in the current model of
the selection mechanism. First, the selection of papers for publication has substantial
impacts on the structure of collaboration networks of authors by determining who and
whose collaboration links can produce acceptable papers and thus be added to the
collaboration networks, which will be further explored in the rest of this chapter. Second,
the current model simply assumes that individual authors would collaborate with anyone
who is readily available, perhaps their previous partners, their students, or even strangers,
rather than choosing those who possess particular skills such that would increase their
chances of getting published. Instead, the success of their collaboration and thus the
73
formation of collaborative connections is determined by the selection criteria of a journal.
In other words, the choice of collaboration partners is blind to the success in publication.
In reality, however, many authors would purposively and strategically choose their
research partners to maximize their chances of acceptance for publication. If that is the
case, the decisions made by individual authors collectively alter the structure of
collaboration networks. In that case, individual authors would need the two different
kinds of information to make rational decisions: (1) the information about a given
journal’s selection criteria to determine which knowledge elements should be included in
their papers, and (2) the information about knowledge elements that all the available
authors possess (i.e., who knows what) to select the best partners to maximize the
likelihood that they can generate acceptable papers through collaboration. The current
approach does not necessarily exclude such possibilities but instead shows that the
observed changes in collaboration networks can be explained by an evolutionary
selection mechanism that does not require individuals to act strategically or rationally.
3.3.2. The Impacts of Selection Criteria at the Network Level
The simple logic of the dyad formation by the selection mechanism can be
extended to more complex structural properties of collaboration networks, when more
than two authors are involved. To explore the impacts of the selection mechanism on the
structure of collaboration networks, a set of simulations were carried out. First, 200
simulated authors were generated, each of who was assigned a unique set of knowledge
elements at random (three elements on each dimension). Then, 1,000 teams were formed,
each of which was composed of one to three authors who were chosen from the pool of
the 200 authors uniformly at random. In that case, the size of teams was determined
74
between 1 and 3 uniformly at random. Finally, a total of 10,000 possible papers were
generated by combining the knowledge elements that individuals or teams possessed. The
generated papers are presented as points on a two-dimensional plane in Figure 3.6. The
points are uniformly distributed over the plane. The white dots represent possible
combinations by single authors, while the grey dots represent those by teams.
Figure 3.6. The possible combinations of knowledge elements by 200 simulated authors
either individually (the white dots) or in teams (the grey dots) and the sweet spots of three
hypothetical journals (the squares).
Next, imagine three hypothetical journals, each of which has a different size of
sweet spot. The first journal has a sweet spot of the smallest area (Journal D; the blue
square on the bottom left in Figure 3.6). Its area is one sixteenth of the total area of the
plane. The second journal has a sweet spot whose area is one fourth of the total area
(Journal E; the green square on the top right). The third journal has the largest sweet spot
that covers the entire plane, meaning that any papers can be selected for publication
75
(Journal F; the red square). Each of three journals selects 300 papers from those within
their sweet spots uniformly at random.
Notice that the strength of selection pressure of a journal is inversely proportional
to the squared area of its sweet spot: that is,
Strength of Selection Pressure ∝
1
(Relative Area of Sweet Spot)
2
.
To demonstrate, compare between Journal D and Journal E. In the case of Journal D all
the possible papers can be divided into two groups by their knowledge elements. One
group is all the papers that fall into the sweet spot of the journal (i.e., acceptable papers),
and the other group is all the other papers (i.e., unacceptable papers). The relative size of
the first group is equal to the relative area of the sweet spot of Journal D (i.e., 1 / 16 =
625 / 10000), because the generated papers are uniformly distributed on the plane.
Probabilistically, one sixteenth of all the papers or about 625 papers will belong to the
first group. Among those 625 papers 300 papers will be submitted and accepted for
publication. Therefore, the expected selection rate of the first group is .48 (= 300 / 625).
On the other hand, 9,375 of the 10,000 randomly generated papers will belong to the
second group, and none of them will be selected for publication. Thus, the selection rate
of the second group is zero (= 0 / 9375). The difference in selection rate between the two
groups is .48 (= |.48 - .00|). In a similar way, the 10,000 randomly generated papers can
be divided into two groups by the characteristics favored by Journal E. Probabilistically,
2,500 papers (= 10000/4) will belong to the first group, and 300 of them will be selected
for publication. In that case, the selection rate of the first group is .12 (= 300 / 2500). On
the other hand, none of papers in the second group will be selected for publication. Thus,
76
the selection rate of the second group is zero (= 0 / 7500). The difference in selection rate
between the first and second groups is .12. Table 3.1 summarizes the comparison
between the two journals.
Table 3.1. The relationship between relative area of “sweet spot” and selection pressure.
Journal D
(Relative area of sweet spot: 1/16)
Journal E
(Relative area of sweet spot: 1/4)
Within
sweet spot
Out of
sweet spot
Total
Within
sweet Spot
Out of
sweet Spot
Total
Selected 300 0 300 Selected 300 0 300
Unselected 325 9,375 9,700 Unselected 2,200 7,500 9,700
Total 625 9,375 10,000 Total 2,500 7,500 10,000
Selection
rate
48%
(300/625)
0%
(0/9375)
Selection
rate
12%
(300/2500)
0%
(0/7500)
Variance in selection rate: .0576 Variance in selection rate: .0036
According to Price’s (1995) general model of selection as well as Fisher’s (1930)
theorem of natural selection, the variance in selection rate between the groups of papers
indicates the strength of selection pressure. The variance in selection rate between the
two groups classified by Journal D’s sweet spot is .0576, and that between the two groups
classified by Journal E’s sweet spot is .0036. This implies that Journal D has 16 times
stronger selection pressure than Journal E’s (.0576 / .0036), while Journal E’s sweet spot
is four times larger than Journal D’s.
Figure 3.7 illustrates the relationship between the area of sweet spot and the
strength of its selection pressure. The bottoms of the pillars represent the sweet spots of
the two hypothetical journals, their heights indicate the probability that a paper within the
77
sweet spot will be actually selected, and the difference in selection rate between two
groups of papers (i.e., the strength of selection pressure)
(a) Journal D
(b) Journal E
Figure 3.7. The relationship between relative area of “sweet spot” and selection pressure.
Technically speaking, the pillars in Figure 3.7 represent bivariate probability
density functions 𝑓 (𝑥 ,𝑦 ) that describe the likelihood of a combination of knowledge
elements [𝑥 ,𝑦 ] to be selected. By Kolmogorov’s (1950) axiom the integral of such a
function over all the plane (i.e., sample space) should be unity:
∬𝑓 (𝑥 ,𝑦 )𝑑𝑥𝑑𝑦 = 1,
which is equivalent to the volume of the pillars. Since its volume is fixed as 1, a pillar
should be tall (i.e., strong selection pressure) when its bottom is narrow (Figure 3.7a) and
it should be short (i.e., weak selection pressure) when its bottom is broad (Figure 3.7b).
Thus, the strength of selection pressure is inversely proportional to the area of its sweet
spot. Further, it should be noted that the location or shape of sweet spots does not matter
at all for measuring the strength of selection pressure, because the relationship between
the area of sweet spots and the height of the corresponding pillars is invariant regardless
of their location or size. In addition, the relationship is also valid when the length of the
78
sequences of knowledge elements is greater than 2. In that case, a hyper-pillar will
represent a multivariate probability density function 𝑓 (𝑥 1
,𝑥 2
,…,𝑥 ℎ
), whose integral is
always equal to 1.
For each journal, the collaboration links were extracted from the authors whose
papers were selected, and a collaboration network was constructed. Figure 3.8 presents
the three collaboration networks and summarizes their structural properties. The relative
area of sweet spot, more precisely, the strength of selection pressure had significant
impacts on the structure of the collaboration networks. Collaboration networks under
stronger selection tend to be better connected and more cohesive than those under weaker
selection. In the collaboration network of Journal D, which is under the strongest
selection among the three, every node is connected to every other node within a few steps
(average path length = 3.62). Further, the network is self-organized into a single large
connected component or so-called “invisible college” (D., J de Solla Price & Beaver,
1966). On the other hand, the collaboration network of Journal F with no substantial
selection pressure is highly spare and fragmented. The network consists of 49
disconnected components, including 34 isolates.
79
(a) Journal D (b) Journal E (c) Journal F
Collaboration
Network
Area of sweet spot Small (1/16) Moderate (1/4) Large (1)
No. nodes (N) 104 137 140
No. edges (E) 296 206 109
No. connected
components
1 9 49
Figure 3.8. Simulated collaboration networks under different levels of selection pressure.
3.4. The Relationship between Selection Pressure and Network Structure
In Section 3.2 the selection criteria of a journal were defined as the collection of
all acceptable papers, denoted by a two-dimensional object having an area, and called the
journal’s sweet spot. Section 3.3 showed that the area of sweet spot is inversely related to
the strength of selection pressure according to Price’s (1995) model of selection. Further,
it suggested that collaboration networks would exhibit different structural properties
under different levels of selection pressure. More specifically, the collaboration networks
tend to be better connected and more cohesive under strong selection than under weak
selection (Figure 3.8).
This section further explores the structural properties of collaboration networks
generated under different levels of selection pressure by carrying out further simulations.
At this time, 20,000 collaboration networks were generated under fined-grained levels of
selection pressure. The area of sweet spot was set to range from 1/10
5
to 1. The structural
80
properties of the 20,000 simulated networks were quantified in two ways. First, the basic
properties of networks were measured by the numbers of (1) nodes and (2) edges, (3)
density, (4) transitivity, and (5) average path length. Next, the spectral properties of
networks—(6) leading eigenvalue, (7) spectral gap, and (8) modularity—were measured
to examine the underlying structure of the networks, such as robustness and community
structure. Detailed explanations about the spectral properties, including their definitions
and implications, are provided in Section 3.4.2.
3.4.1. The Basic Properties of the Collaboration Networks
First, the simulation results suggested that collaboration networks under strong
selection pressure tended to contain fewer nodes than those under weak selection
pressure. In Figure 3.9a the number of nodes is negatively related to the strength of
selection pressure on a double logarithmic scale. The results are intuitive. Nodes in
collaboration networks represent authors who can contribute to producing acceptable
papers by providing their knowledge elements during collaboration. Such authors would
be rare under strong selection, because strong selection implies a small sweet spot or a
narrow range of acceptable knowledge elements and that few authors possess such
elements. Therefore, fewer authors can produce acceptable papers and join the network
under strong selection than under weak selection.
Second, the relationship between the number of edges and selection pressure was
found to be an inverted U-shaped curve (Figure 3.9b). As selection pressure became
stronger, collaboration networks tended to have more edges and became better connected.
Under strong selection, only few authors possess produce acceptable papers by
themselves. But some other authors could produce acceptable papers through
81
collaboration, although they were unable to produce acceptable papers by themselves.
Thus, it is intuitive that more edges were found under strong selection. However, the
number of edges stopped increasing and began to decrease at a moderate level of
selection pressure (i.e., the blue vertical line in Figure 3.9b). This is largely because of
the decrease in the number of nodes between which edges could be formed.
(a)
(b)
Figure 3.9. The numbers of (a) nodes and (b) edges of the 20,000 simulated collaboration
networks. Note: the solid red lines indicate the medians, and the dashed red line indicates
the 95% confidence intervals.
Next, the density and transitivity of the simulated collaboration networks
increased in very similar patterns, as the selection pressure became stronger (Figure
3.10). Collaboration networks were better connected and more clustered under strong
selection than under weak selection, which confirms the observed tendency in the
previous section. Fewer authors could produce acceptable papers under strong selection
than weak selection, but an equal number of papers were produced regardless of the
strength of selection pressure. This means that repeated authors and collaborations were
82
more frequent, and therefore, collaboration networks were denser and more clustered
under strong selection than under weak selection.
(a)
(b)
Figure 3.10. (a) Network density and (b) transitivity of the 20,000 simulated
collaboration networks. Note: the solid red lines indicate the medians and the dashed red
lines indicate the 95% confidence intervals.
Finally, the average path length of the simulated networks were examined. In that
case, the length of the missing paths were measured as the number of nodes plus one to
penalize disconnected networks (West, 2001). As shown in Figure 3.11, average path
length was negatively correlated to the strength of selection pressure. This means that
nodes are more closely connected under strong selection than under weak selection
pressure.
Overall, the examination of the basic properties of the simulated collaboration
networks confirmed that the observed tendency that collaboration networks are better
connected under strong selection than under weak.
83
Figure 3.11. Average path length of the 20,000 simulated collaboration networks. Note:
the solid red line indicates the median, and the dashed red lines indicate the 95%
confidence intervals.
3.4.2. The Spectral Properties of the Collaboration Networks
In graph theory, the spectrum of a network refers to a listing of the eigenvalues of
the adjacency matrix of the network (Chung, 1997; Diestel, 2006; Steen, 2010).
Undirected networks, in which edges have no orientations as in collaboration networks,
can be represented by symmetric adjacency matrices whose elements are either 0 or 1.
Any 𝑛 × 𝑛 real symmetric matrix has n real eigenvalues, 𝜆 1
≥ 𝜆 2
≥ ⋯ ≥ 𝜆 𝑛 and an equal
number of orthonormal eigenvectors, 𝛾 1
⃗⃗⃗ , 𝛾 2
⃗⃗⃗ , …, 𝛾 𝑛 ⃗⃗⃗ associated with those eigenvalues.
The eigenvalues of adjacency matrices provide important information about the
underlying structure of networks, called spectral properties. In particular, the largest
eigenvalue, denoted by 𝜆 1
, and the spectral gap, the difference between the first and
second largest eigenvalues, denoted by Δ𝜆 (= 𝜆 1
− 𝜆 2
), contain useful information about
the connectedness of networks, more precisely, network robustness.
84
There are two different approaches to assessing network connectedness. The first
and straightforward way is to measure the fraction of the pairs of connected nodes out of
all possible pairs (i.e., density), the fraction of closed triads of all connected triads (i.e.,
transitivity), or the average number of steps to get one node to another (i.e., average path
length). Alternatively, network connectedness can be assessed by asking the minimum
number of nodes or edges that need to be removed to disconnect the remaining nodes
from each other (i.e., node- or edge-connectivity) or whether a network remains
connected after a certain number of nodes (or edges) are removed (Diestel, 2006). The
better connected a network is, the more nodes (or edges) need to be removed to render
the network disconnected, and the more likely it will remain connected after a given
number of nodes (or edges) are removed. This idea is closely related to the concept of
network robustness.
Network robustness is defined as the capacity of networks to remain functional (in
terms of connectivity) in the face of perturbations, such as the random or targeted
deletion of nodes or edges (Bollobás & Riordan, 2004; Callaway, Newman, Strogatz, &
Watts, 2000; Greenbury, Johnston, Smith, Doye, & Louis, 2010). Network robustness is a
very useful concept for studying the reliability of information flows in communication
networks or traffic flows in transportation networks (Boccaletti et al., 2006). Notice that
the robustness of a network measures how insensitive a network is (or would be) to local
changes that might be caused by exogenous factors instead of how well connected the
network is at a given moment. In this sense, network robustness well captures a dynamic
characteristic of networks, which is important for understanding network evolution
(Demetrius & Manke, 2005).
85
Demetrius and Manke (2005; see also Manke, Demetrius, & Vingron, 2006)
proposed network entropy as an analytical characterization of network robustness. Based
on the concept of Kolmogorov-Sinai entropy (Kolmogorov, 1959; Sinai, 1959), which is
a generalization of Shannon’s (1948) entropy and describes the rate at which a stochastic
process generates information, the authors formulated information flow in a network as a
sequence of nodes visited by a Markov process on the network. A Markov process (also
called “Markov chain”) is a stochastic model used to study a random system that changes
states according to a transition rule. The transition rule is defined as a set of conditional
probabilities of the next states given the current states and is usually denoted by a square
matrix 𝐏 = { 𝑝 }
𝑖𝑗
= 𝑃𝑟𝑜𝑏 (state 𝑗 |state 𝑖 ). The row sum of the matrix ∑𝑝 𝑖𝑗 𝑗 is always
equal to 1, and such a matrix is called a stochastic matrix or a Markov matrix (Terrell,
1999). Demetrius and Manke (2005) defined network entropy H(P) as the entropy of a
stochastic matrix associated with the adjacency matrix of the network:
𝐻 (𝐏 ) = ∑𝜋 𝑖 𝐻 𝑖 𝑛 𝑖 =1
,
where 𝜋 is the stationary distribution of the Markov matrix P or its left-eigenvector (i.e.,
𝜋 = 𝜋 𝐏 ), and 𝐻 𝑖 is the Shannon entropy ∑𝑝 𝑖𝑗
log (𝑝 𝑖𝑗
)
𝑗 . Further, the authors showed that
in the case of undirected networks network entropy is equal to the logarithm of the largest
eigenvalue of the adjacency matrix: that is,
𝐻 (𝐏 ) = log𝜆 1
.
Figure 3.12 illustrates the validity of the largest eigenvalue 𝜆 1
as a reliable
measure of network robustness by comparing four networks. Those networks have the
86
same numbers of nodes (N = 100) and edges (E = 99) but different values of 𝜆 1
owing to
their different structures. Among the four networks, the line network is the most
vulnerable to random removal of nodes and/or edges (Figure 3.12a). Expect for the two
end nodes, the removal of any single node will divide the network into two disconnected
parts. This means that the network will be disconnected by random removal with a
probability of 98%. On the other hand, the star network is the least sensitive to random
removal (Figure 3.12d). It can remain connected unless the central node is removed,
which will happen with probability of 1%. Therefore, the star network is better connected
than the line network. The largest eigenvalue of the adjacency matrix of a network well
reflects the ability of that network to remain connected under the random removal of
nodes. The largest eigenvalue of the line network is 1.999, while that of the star network
is 9.950.
(a) Line network
λ1 = 1.999
(b) Small-world
network
λ1 = 2.683
(c) Scale-free
network
λ1 = 4.393
(d) Star network
λ1 = 9.950
Figure 3.12. The largest eigenvalue as a measure of network robustness against random
failure. Note: Each network has the same numbers of nodes (N = 100) and edges (E =
99).
When nodes are selected for removal at random (i.e., random failure), the star
network is the most reliable and robust. However, if the selection is made deliberately
87
(i.e., targeted attacks), it is the most vulnerable. The removal of a single node (i.e., the
hub) will lead to an immediate breakdown of the whole network. The other three
networks, on the other hand, will remain connected, at least partially, even after the most
central nodes are removed. From this perspective the star network may not be the best
connected one. In general, it is known that networks with homogenous connectivity (i.e.,
decentralized networks) are robust against targeted attacks (Albert, Jeong, & Barabási,
2000; Crucitti, Latora, Marchiori, & Rapisarda, 2004)
Estrada (2006) proposed the difference between the first and second largest
eigenvalues of the adjacency matrix of a network as an indicator of the robustness of the
network against targeted attacks. The difference is called “spectral gap” and denoted by
Δ𝜆 (= 𝜆 1
− 𝜆 2
). Figure 3.13 presents four different networks with different levels of
robustness against targeted attacks and their spectral gaps. The four networks are in
common composed of two clusters of an equal size: one is the red nodes, and the other is
the blue nodes. But the numbers of edges connecting the two clusters or bridges differs
(the green lines in Figure 3.13). Because the removal of the bridges or their nodes will
make the networks disconnected, the different numbers of bridges represent the different
levels of robustness against targeted attacks. As shown in Figure 3.13, the networks with
more bridges are more robust and have larger spectral gaps.
Using the largest eigenvalue and spectral gap of networks, the connectedness of
the 20,000 simulated collaboration networks were examined in relation to the strength of
selection pressure. The simulation results suggested that under strong selection
collaboration networks tended to have high levels of 𝜆 1
but low levels of Δ𝜆 (Figure
88
3.14). In other words, collaboration networks under strong selection were robust against
random errors but vulnerable to targeted attacks.
No. bridges 0 1 5 10
λ1 4 4.24 5 6.09
λ2 4 3.83 3 2.21
Δλ 0 0.41 2 3.88
Figure 3.13. Spectral gap as a measure of network robustness against “targeted” attacks.
(a)
(b)
Figure 3.14. (a) The largest eigenvalue and (b) spectral gap of the 20,000 simulated
collaboration networks. Note: the solid red lines indicate the medians and the dashed red
lines indicate the 95% confidence interval.
89
Figure 3.15 presents one of the simulated collaboration networks generated under
strong selection. Of the 200 available authors, only 42 authors could produce acceptable
papers and thus join the network. The collaboration network was fairly dense and
clustered (density = .148; clustering coefficient = .446), and the 42 nodes were connected
by short paths (average path length = 2.566). Overall, the network was fairly well
connected. All these results are consistent with the results discussed in Section 3.4.1. In
addition, the network was expected to be highly robust against random errors (λ1 =
48.74). However, the network was vulnerable to targeted attacks, exhibiting a relatively
low level of spectral gap (Δλ = 9.14). The network consisted of four cohesive clusters in
which nodes were densely connected to each other, but the clusters were loosely
connected by a few bridges (the green lines in Figure 3.15). The removal of those bridges
or their end points (i.e., targeted attacks) will render the network disconnected. The fact
that the network was vulnerable to targeted attacks implies that the network was poorly
connected, which appears to be logically incompatible with the other findings.
No. nodes: 42
No. edges: 127
Density: .148
Clustering coefficient, C: .446
Average path length, L: 2.566
No. bridges: 7
λ1 = 48.74, λ2 = 39.60, Δλ = 9.14
Modularity, Q: .578
Figure 3.15. A simulated collaboration network under strong selection (the area of the
sweet spot is 1/1000 of the total area).
90
The concept of modularity helps to resolve the logical contradiction observed in
the simulation results. Modularity is referred to as the tendency for the components of
complex systems to be organized into semi-independent groups, that is, those that are
loosely related to or little interdependent on one another (Bastolla & Parisi, 1997;
Callebaut & Rasskin-Gutman, 2005; Clune, Mouret, & Lipson, 2013; Holme, 2011;
Yoon, Blumer, & Lee, 2006). The concept of modularity is crucial to understand the
diversification within complex systems and the interaction between the systems and their
environments in evolutionary processes (Clune et al., 2013; Hansen, 2003). Previous
research has found that modular systems abound both in natural and artificial worlds
because of their evolutionary advantages (Clune et al., 2013). Herbert Simon (1996)
introduced the famous parable of two watchmakers to explain the evolutionary
advantages of modular systems:
There once were two watchmakers, named Hora and Tempus, who made very fine
watches. The phones in their workshops rang frequently and new customers were
constantly calling them. However, Hora prospered while Tempus became poorer
and poorer. In the end, Tempus lost his shop. What was the reason behind this?
The watches consisted of about 1000 parts each. The watches that Tempus
made were designed such that, when he had to put down a partly assembled
watch, it immediately fell into pieces and had to be reassembled from the basic
elements. Hora had designed his watches so that he could put together
subassemblies of about ten components each, and each sub-assembly could be put
down without falling apart. Ten of these subassemblies could be put together to
91
make a larger subassembly, and ten of the larger subassemblies constituted the
whole watch (p. 188).
Notice that the watches that Hora made were no less complex than those of Tempus. But
Hora decomposed the original task into smaller subtasks so that the failure of one subtask
influenced little the completion of other subtasks or the whole task. Therefore, Hora
could build more watches than Tempus under the same condition of interruption.
10
In network research, a network is said to be modular if nodes in the network are
densely connected to one another within their clusters (or modules) but relatively
disconnected to the others (Newman, 2006). The concept of network modularity has been
most often adopted for detecting and characterizing the community structure of empirical
networks (Clauset, Newman, & Moore, 2004; Girvan & Newman, 2002). The community
structure of collaboration networks can convey useful information. If a collaboration
network of a journal is found to be highly modular so that it is composed of several
subsets of densely connected nodes, corresponding to groups of researchers who have
worked closely together, it is reasonable to conceive that different groups of researchers
are interested and specialized in different research topics, implying the emergence of
subareas within the journal.
However, the concept of network modularity alone does not account for why
collaboration networks are more modular under strong selection than weak selection. In
other words, what are the evolutionary advantages of modular networks? A recent study
10
Note that Hora did not just decompose his tasks into subtasks but further decomposed those subtasks into
even smaller pieces. When a complex system can be decomposed at multiple levels, as Hora did, the
system is said to be hierarchically modular, or simply, hierarchic, which is known as the most survivable
and thus common form of complex systems in unstable environments and is a key mechanism by which the
complex forms possibly arise from the simple ones by purely random processes (Simon, 1962).
92
by Clune, Mouret, and Lipson (2013) provides an explanation for the evolutionary
advantages of modular networks under strong selection. According to them, network
modularity is a byproduct of the dual selection pressure to maximize network
connectedness and to minimize connection costs of networks. Connection costs include
creating connections, maintaining them, the effort to transmit information along them,
and the delays in transmission, all of which increases as a function of the number of
edges in a network. Densely connected networks facilitate information flows in
communication networks. However, it is costly to build and maintain such networks. In
the context of scientific collaboration networks, connections represent collaborations
between researchers and entail nontrivial costs, including the communication costs for
geographically distributed teams (J. Katz, 1994; Luukkonen, Persson, & Sivertsen, 1992),
the administrative burdens in inter-institutional collaborations (Landry & Amara, 1998),
the coordination costs for culturally diverse teams (Bercovitz & Feldman, 2011).
Individual authors need to be efficiently connected to each other so that they can reduce
collaboration costs. Therefore, collaboration networks are expected to be more modular
under strong selection.
The final part of the simulation results confirmed this expectation. The modularity
of the 20,000 simulated collaboration networks were measured following the
computational algorithm suggested by Clauset, Newman, and Moore (2004). Network
modularity is defined as the fraction of the edges that fall within the given clusters minus
the expected fraction if edges were distributed at random. The value of the modularity
ranges from -.50 to 1.00. It is positive if the number of edges within clusters exceeds the
number expected on the basis of chance. According to this formula, however, networks
93
with many isolated clusters would also show high levels of modularity. To avoid this
confusion, the modularity of the largest connected components of the simulated networks
was measured. As expected, the results showed that collaboration networks are more
modular under strong selection than under weak selection (Figure 3.16).
Figure 3.16. The modularity of the 20,000 simulated collaboration networks (the largest
connected component only). Note: the solid red line indicates the median and the dashed
red lines indicate the 95% confidence interval.
3.5. Summary
The publication of academic papers alters the structure of scientific collaboration
networks, because the authors of the published papers and their collaborative connections
are newly added to existing networks as nodes and edges. Not all papers, however, are
submitted or accepted for publication. Only a few of them are selected. Depending on
which papers are selected for publication, collaboration networks change in different
ways. Therefore, the course of the evolution of collaboration networks is determined by
the selection of papers for publication.
94
The mechanism of the selection of papers can be formulated based on the Price’s
(1995) general model of selection. The fact that papers are selected by certain mechanism
means that they are selected at different rates (i.e., 𝑤 𝑖 ≠ 𝑤 𝑗 ) because of their different
characteristics (i.e., 𝑧 𝑖 ≠ 𝑧 𝑗 ), namely, the different combinations of knowledge elements
that constitute the papers. When it is possible to measure various knowledge elements of
the papers and compare their selection rates, the relationship between selection rate and
paper characteristics 𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 ) can be found, which allows us to estimate the direction
and intensity of selection pressure on papers. Further, the relationship will make it
possible to predict the direction of the evolution of collaboration networks. However, it is
extremely difficult, if possible at all, to empirically measure the knowledge elements of
all the papers in an objective way, including the accepted and rejected papers and even
those that are not submitted yet.
Alternatively, the current model of selection mechanism focuses on the strength
of selection [i.e., the range of papers preferred, 𝑣𝑎𝑟 (𝑤 𝑖 )] rather than the direction of
selection [i.e., the kind of papers preferred, 𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )]. For this reason, the current
model does not directly address the question about what kind of papers are likely to be
selected. Nonetheless, it allows us to predict the structural properties of collaboration
networks that would develop under certain levels of selection pressure. More specifically,
assuming the model of paper generation utilized under strong selection (1) collaboration
networks tended to exhibit high levels of the largest eigenvalue 𝜆 1
, implying that they
were robust against random failures (i.e., the random deletion of nodes or edges),
whereas (2) they showed small spectral gaps Δ𝜆 , implying that they were vulnerable to
targeted attacks (i.e., the deliberate damage on central nodes and edges). It seems to be
95
logically incompatible that collaboration networks are well-connected in one respect but
poorly connected in another simultaneously. However, these observations together
suggested that collaboration networks are expected to evolve into ones that maximize
network connectedness and minimize connection costs at the same time under strong
selection. In other words, collaboration networks would be (3) self-organized into
modular networks that are composed of several cohesive clusters of nodes, which
increase and maintain the high levels of connectedness among nodes, but those clusters
are loosely connected to one another, which reduces connection costs.
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CHAPTER 4: NETWORK GROWTH BY REPLICATION AND MUTATION
The previous chapter proposed a model of the selection mechanism in which
papers with specific knowledge elements are more likely than others to be submitted and
accepted for publication, and thus, authors who are capable to produce such papers either
by themselves or through collaboration can join collaboration networks. Following
Price’s (1995) general model of selection, the range of acceptable papers was interpreted
as the strength of selection pressures and expected to have significant impacts on the
structural properties of collaboration networks. When the range of acceptable papers is
narrow (i.e., under strong selection), only a small subset of authors can produce
acceptable papers. As the particular authors and their collaborative connections are added
to an existing collaboration network, the structure of the collaboration networks is
expected to change in different ways than they would change if any authors and their
collaborative connections are added. More specifically, the simulation results in the
previous chapter suggested that collaboration networks tend to show high levels of the
largest eigenvalue and modularity but small spectral gaps under strong selection.
It should be noted, however, that collaboration networks will grow and change
even when any authors and their collaborative connections are added regardless of their
ability to produce acceptable papers. Put differently, the structural change of
collaboration networks is expected even in the complete absence of selection pressure as
far as new authors and their collaborative connections are continuously added. Further,
the collaboration networks could happen to change in the direction predicted by the
model of the selection mechanism. If that is the case, the observed structural change
should not be considered as a result of the selection mechanism. But instead, it might be
97
simply due to the growth of collaboration networks. Therefore, the effects of the selection
mechanism on the structural change of collaboration networks should be teased apart
from those of the growth of the networks over time.
From this perspective this chapter proposes a model in which collaboration
networks grow over time as authors and their collaborative connections are added and
explores the structural change of collaboration networks due to the simple growth
mechanism. In this dissertation, the growth of collaboration networks refers to the
addition of sets of authors and their collaboration links, which are called “authoring
teams.” An authoring team is a group of authors working together to produce a journal
article and is the smallest unit that can possibly (but not necessarily) alter the structure of
collaboration networks. Although the word team is commonly used to refer to a group of
people as opposed to a single individual, authors who produce papers by themselves will
be also considered as authoring teams, in particular, teams with one member each. In
collaboration networks, authoring teams are represented by subsets of nodes. An
authoring team is represented by a single node when the team has only one member, by a
connected dyad when the team has two members, and by a clique, which every node is
connected with every other node (Wasserman & Faust, 1994), when the team is
composed of more than two members.
The basic logic of network growth is based on the other two evolutionary
mechanisms—replication and mutation. When a team of authors work together to publish
a paper, an authoring team is formed. The team will be replicated if the authors repeat
their collaboration to publish new papers. However, the replication may or may not be
perfect. Mutation occurs when some authors leave the team, and/or when new authors
98
join the team. The replication of existing authoring teams will maintain the structural
patterns of an existing network, whereas the mutation will give rise to variant patterns in
the network, which may lead to the structural change of the network. The current growth
model (1) describes the process in which new authoring teams are formed by the
replication of existing teams with mutation and (2) generates possible networks into
which a collaboration network observed at a point in time would change at the very next
point in time by adding the newly formed authoring teams. The structural properties of
those possible networks as a whole will be interpreted as the expected structure that a
collaboration network would exhibit if it grows in the absence of selection pressures. The
current model is based on the self-organization model of bipartite networks (Ramasco et
al., 2004; Ramasco & Morris, 2006) and the vertex copying process (Kleinberg et al.,
1999; Kumar et al., 2000, 1999).
4.1. Bipartite Collaboration Networks
The current model defines a collaboration network at a given moment as a
bipartite network and describes the process by which the bipartite network grows as the
new columns and rows are added over time. This section will introduce the basic
concepts and measures that characterize bipartite collaboration networks, which will be
used as the parameters of the growth model in the following section. The notations used
in this chapter are based on Newman (2010).
In most previous studies, scientific collaboration networks are defined by a set of
authors and by another set of undirected (and often unweighted) edges among the
authors. A pair of authors are connected by an edge if they have written a paper together.
Usually, collaboration networks of authors are obtained by the one-mode projection of
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bipartite networks (e.g., Ding, 2011; Guimerà et al., 2005; Newman, 2001c; Sarigöl,
Pfitzner, Scholtes, Garas, & Schweitzer, 2014; Yan & Ding, 2009). A bipartite network
(also called an “affiliation network,” Wasserman & Faust, 1994) consists of two disjoint
sets of nodes (e.g., authors and papers) and the connections from the nodes of one set to
those of the other set. However, no connections between nodes of the same set are
allowed in a bipartite network. One-mode collaboration networks are often dichotomized,
so that a pair of authors are connected if they have written at least one paper together
regardless of how many papers they have worked on together.
One-mode projections necessarily lead to information loss. One-mode networks
of authors cannot specify which paper a pair of connected authors have worked on, which
is a critical piece of information for the current model. Also, dichotomization leads to
further information loss. Dichotomized one-mode collaboration networks do not provide
the information about how many papers a pair of connected authors have worked on
together, which can be interpreted to indicate the tendency that authors repeat their
collaboration, namely, relational inertia (Ramasco & Morris, 2006). For these reasons,
the evolutionary process of collaboration networks can be better understood by
examining bipartite networks of collaborations rather than one-mode projected networks
of authors.
A bipartite network of collaborations at a given point in time is represented by a
(𝑁 𝑎 × 𝑁 𝑝 ) matrix B, where 𝑁 𝑎 is the number of authors and 𝑁 𝑝 is the number of
published papers (Figure 4.1). Each element of matrix B, 𝑏 𝑖𝑘
, is 1, if author i wrote paper
k; and 𝑏 𝑖𝑘
is 0, otherwise. The kth column of matrix B, [𝑏 1𝑘 ,𝑏 2𝑘 ,…,𝑏 𝑁 𝑎 𝑗 ]
𝑇 , indicates all
the authors who worked as a team on paper k, that is, the kth authoring team. Its sum
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∑𝑏 𝑖𝑘 𝑖 represents the total number of authors of paper k, denoted by 𝑛 𝑘 . On the other
hand, the ith row of matrix B, [𝑏 𝑖 1
,𝑏 𝑖 2
,…,𝑏 𝑖 𝑁 𝑝 ], indicates all the published papers of
author i, and its sum ∑𝑏 𝑖𝑘 𝑗 represents the total number of the published papers of author i
or simply author i’s publication experience, denoted by 𝑞 𝑖 . Given that a bipartite network
of collaborations is constructed from raw data, two frequency distributions can be
immediately obtained. One is 𝑃 𝑛 (𝑛 ) that describes the frequency of the number of authors
per paper; the other is 𝑃 𝑞 (𝑞 ) that represents the frequency of the number of publications
per author.
Figure 4.1. A bipartite network of collaborations and its growth.
The post-multiplication of matrix 𝐁 by its transpose 𝐁 𝑇 returns the one-mode
undirected weighted networks of authors, 𝐖 (𝑁 𝑎 ×𝑁 𝑎 )
, whose diagonal element 𝑤 𝑖𝑖
is the
number of papers that author i has written, which is always equal to author i’s experience
𝑞 𝑖 (i.e., 𝑤 𝑖𝑖
= ∑𝑏 𝑖𝑗
2
𝑗 = ∑𝑏 𝑖𝑗 𝑗 = 𝑞 𝑖 , because 𝑏 𝑖𝑗
∈ { 0,1}). On the other hand, the off-
101
diagonal element 𝑤 𝑖𝑗
is the number of papers that author i and author j have coauthored
(i.e., 𝑤 𝑖𝑗
= ∑𝑏 𝑖𝑘
𝑏 𝑗𝑘 𝑘 ) and can be interpreted to indicate the strength of the collaborative
relationship between authors i and j. The ith row sum ∑ 𝑤 𝑖𝑗 𝑗 ≠𝑖 indicates authors i’s
overall strength of collaborative relationship and is denoted by 𝑠 𝑖 . The unweighted
adjacent matrix of authors A can be obtained by dichotomizing W with a cutoff value of
1 and by excluding the diagonal elements. Specifically, the element of matrix A, 𝑎 𝑖𝑗
= 1,
if 𝑤 𝑖𝑗
≥ 1 for all 𝑖 ≠ 𝑗 : that is, if authors i and j have worked together on one or more
papers; otherwise, 𝑎 𝑖𝑗
= 0. The ith row sum ∑𝑎 𝑖𝑗 𝑖 indicates the total number of the
distinctive collaborators of authors i (i.e., the degree of node i in a one-mode
collaboration network of authors) and is denoted by 𝑘 𝑖 . The frequency distribution 𝑃 𝑘 (𝑘 )
describes the number of authors who have k collaborators, or more simply, the degree
distribution of the one-mode collaboration network.
Now consider the tendency that individual authors repeat collaboration with their
previous partners. Let Ω
𝑖 denote the ratio of the total number of papers that author i has
published through collaboration 𝑠 𝑖 to the total number of distinctive collaborators of the
author 𝑘 𝑖 :
Ω
𝑖 =
𝑠 𝑖 𝑘 𝑖 .
The ratio Ω
𝑖 ranges from 1 in the cases where authors never repeat collaborations (i.e.,
𝑠 𝑖 = 𝑘 𝑖 ) to 𝑞 𝑖 in the cases where authors always write papers with the same collaborators
(i.e., 𝑠 𝑖 = 𝑞 𝑖 𝑘 𝑖 ). A large Ω
𝑖 suggests that authors tend to repeat collaborations with their
former collaborators rather than finding new partners. Further, define 𝑟 𝑖 as the rate at
which authors i repeats their collaborations with their former collaborators:
102
𝑟 𝑖 = 1 −
1
Ω
𝑖 = 1 −
𝑘 𝑖 𝑠 𝑖 .
The rate of repeated collaboration 𝑟 𝑖 is 0, if author i has never repeated collaborations
(i.e., Ω
𝑖 = 1), and approaches 1, if author i has written a large number of papers with the
same partners (i.e., Ω
𝑖 = 𝑞 𝑖 ≫ 0). Further, the overall rate at which collaborations are
repeated in a collaboration network can be defined as
𝑅 = 1 −
∑∑𝑎 𝑖𝑗 𝑖 𝑗 ∑∑𝑤 𝑖𝑗 𝑖 𝑗 .
This rate can be called relational inertia and measures the overall tendency that authors
repeat their collaborations with their previous partners.
A bipartite network of collaborations grows over time by the continuous addition
of newly published papers (i.e., new columns) and of new authors with no previous
publication experience (i.e., new rows). Notice that no new authors can join collaboration
networks without a new paper being published. That is, the number of rows of a bipartite
matrix B will increase only when the number of columns increases. On the other hand,
the number of columns of matrix B (i.e., the number of papers) can grow without any
increase in the number of rows (i.e., the number of authors), if newly published papers
are written by experienced authors. Putting these two facts together, it is possible to
propose that the ratio of the total number of authors 𝑁 𝑎 to the total number of published
papers 𝑁 𝑝 indicates the average rate at which new authors participate in new papers at
each point in time, which is denoted by 𝑚 ̅:
𝑚 ̅ =
𝑁 𝑎 𝑁 𝑝 .
103
That is, a new author joins the collaboration network every 𝑚 ̅ time of a new publication.
Put differently, each authoring team has a new author at rate of 1/𝑚 ̅. To demonstrate,
suppose that every single author has written one and only one paper without any
collaboration. Then, the bipartite matrix B is an (𝑁 𝑎 × 𝑁 𝑎 ) identity matrix, whose
diagonal elements are all equal to 1 and off-diagonals are zero. If that is the case,
whenever a new column is added, a new row is also added, meaning that a new author
joins the collaboration network every time a new paper is published. Therefore, the ratio
of 𝑁 𝑎 to the total number of published papers 𝑁 𝑝 (in this case, 1 = 𝑁 𝑎 /𝑁 𝑝 ) well
represents the overall rate at which new authors join the collaboration networks.
However, note that because 𝑚 ̅ is a global-level measure, it may not accurately represent
the introduction rate of new authors to each team of authors who initiates a new project
together. Instead, a conditional probability that the number of new authors is m given the
size of team n, 𝑃 𝑚 |𝑛 (𝑚 |𝑛 ), can provide more detailed information
So far, six measures have been discussed, all of which are obtainable directly
from either the bipartite or projected forms of collaboration networks. These measures
will be used as parameters in the network growth mechanism, which is discussed in the
following section. Table 4.1 summarizes the six parameters.
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Table 4.1. The parameters for the team formation in collaboration networks
Parameter Definition Description Frequency
𝑛 𝑗 ∑𝑏 𝑖𝑗 𝑖 The number of authors of paper j 𝑃 𝑛 (𝑛 )
𝑞 𝑖
∑𝑏 𝑖𝑗 𝑗
The number of published papers by
author i
𝑃 𝑞 (𝑞 )
𝑤 𝑖𝑗
∑𝑏 𝑖𝑘
𝑏 𝑗𝑘 𝑘
The number of published papers on
which authors i and j have worked
together
𝑃 𝑤 (𝑤 )
𝑘 𝑖
∑𝑎 𝑖𝑗 𝑖
The number of collaboration partners
of author i
𝑃 𝑘 (𝑘 )
𝑅 1 −
∑∑ 𝑎 𝑖𝑗 𝑖 𝑗 ∑∑𝑤 𝑖𝑗 𝑖 𝑗 .
The average rate at which authors
repeat collaborations with their former
collaborators
𝑃 𝑟 (𝑟 )
𝑚 ̅
𝑁 𝑎 𝑁 𝑐
The average rate at which a new author
joins an authoring team
𝑃 𝑚 |𝑛 (𝑚 |𝑛 )
4.2. A Growth Model of Collaboration Networks
This section proposes a model that describes the growth of collaboration networks
by the addition of new authoring teams, which are formed by the replication of existing
teams with mutation. The proposed model utilizes the six parameters that were discussed
in the previous section.
To begin, consider a simplistic case in which authoring teams are formed by the
replication mechanism only. Suppose that an existing team k works on a new project and
their paper gets published. In that case, the bipartite matrix B is expanded by the addition
of a new column that is an exact copy of the kth column of B. Notice that the publication
of a new paper by the repeating team increases the total number of papers by one.
105
However, this does not result in substantial structural change in the one-mode
collaboration networks of authors, because neither new authors nor new links have been
added. The only structural change is that the connections among the members of team k
are reinforced, as the new paper increases the weight of each of the connections among
them by one.
Figures 4.2a illustrate a hypothetical scenario in which a team is formed by exact
replication. Up to time t, five authors have written five papers (i.e., the ancestral network
at the top of Figure 4.2). Among these five authors, those who wrote paper b together
(i.e., authors 2, 3, and 4) collaborate again and write another paper b* at time t + 1. In this
scenario, the team members have not changed. Therefore, the number of nodes and
linkage patterns of the collaboration network remain the same. However, the connections
among the three team members have been reinforced (the blue lines in Figure 4.2a).
It is common that a group of researchers work together and write papers
repeatedly. But it is also common that authoring teams often recruit new members,
especially when incumbent members drop out. If the latter is the case, the new column
that is added to the collaboration matrix B is no longer exactly identical to the column
selected to be copied, although the two columns might be very similar. This imperfect
copying of columns is considered as mutation in the current model.
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(a) Ancestral network
(a) Replication (b) Mutation (c) Mutation
Replacement of team
members with experienced
authors
Replacement of team
members with new authors
Figure 4.2. The growth of a collaboration network (a) by exact replication, (b) by
replication with replacement of team members with experienced authors, and (c) by
replication with replacement of team members with new authors.
The members of an authoring team can change in two different ways. First,
experienced authors can be invited as new members to the team, especially when they
have prior experiences of collaboration with the current members. Figure 4.2b illustrates
this possibility. The authoring team who wrote paper b together initiates a new project.
However, author 2 drops out this time. The team may search for a new member who can
undertake the role that author 2 played. Either authors 1 or 5 may be chosen as a new
member. However, it would be reasonable to expect that author 1 is more likely to join
the team than author 5, because author 1 worked with author 4 on paper a before (i.e.,
107
relational inertia) but author 5 is a stranger to all the existing members. In general, those
who have prior experiences of successful collaboration with the existing members of a
team are more likely to join the team (Ramasco & Morris, 2006). If author 1 joins the
team, and if the team successfully produces another paper b*, a new link will be formed
between authors 1 and 3 (the green line in Figure 4.2b) and the existing links between
authors 1 and 4 and between authors 3 and 4 will be reinforced (the blue lines in Figure
4.2c). This kind of membership change will result in moderate structural changes in one-
mode collaboration networks by adding new collaborative relations. Still, the total
number of nodes remains the same.
The second possible way of membership change is that new authors with no
previous publication experiences join a team and replace previous members of that team.
Figure 4.2c illustrates this scenario. Author 6, who has no previous publication
experience, joins the team of authors 3 and 4 and plays the role of author 2 or perhaps
adds new competencies. This kind of membership change yields more salient structural
changes in one-mode collaboration networks than the two previous cases. As author 6
joins the team, the number of nodes increases from five to six (the green circle in Figure
4.2d) and two collaborative connections are newly formed between authors 3 and 6 and
between authors 4 and 6 (the green lines in Figure 4.2d). Although the two types of
membership change—replacing existing members with experienced or inexperienced
authors—are described separately, both can happen within a team at the same time. That
is, some of the incumbent members are replaced by other experienced authors, others are
replaced by new authors with no previous experiences, and the rest of the incumbent
members continue to work in the team.
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Integrating the three possible ways of team formation—(i) exact replication, (ii)
replacement of incumbent members with experienced authors, and (iii) replacement of
incumbent members with inexperienced authors, the growth mechanism of a bipartite
collaboration network is defined by the following rules:
1. At every point in time, an existing authoring team is sampled for replication
uniformly at random.
11
2. Of n members of the selected team, m members are selected at random and
replaced with the same number of members with no previous publication
experiences. Here, m is obtained as a random variable from the conditional
probability distribution 𝑃 𝑚 |𝑛 (𝑚 |𝑛 ).
3. Of the remaining (n – m) members, l members will be selected with probability
proportional to their tendencies of repeated collaboration 𝑟 𝑖 and continue to work
in the team.
4. The remaining (n – m – l) members will drop out of the team and be replaced with
other experienced authors, those who will be selected depending on their prior
experiences of collaboration with the l continuing authors, 𝑤 𝑖𝑗
.
5. The newly formed team is added to the existing network, resulting in the growth
and structural change of the network.
It is important to note that the teams that are formed by the replication and
mutation processes are not groups of “strangers” who are randomly chosen. Because the
11
For the first step different sampling rules can be applied so that recent teams are more likely than old
teams to be sampled for replication. Section 5.6 presents the results from the recent-team-favored sampling
methods.
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five-step team formation mechanism is strictly governed by the six parameters obtained
from empirical data (Table 4.1), generated teams hold the relational patterns that are
statistically similar to those of actually existing teams. For example, the average size of
teams generated by the current mechanism is supposed to be equal to the average size of
actual teams, because the size of generated teams follows 𝑃 𝑛 (𝑛 ), which is the distribution
of the size of actual teams. More generally, the team formation mechanism well reflects
the norms and conventions developed over time in actual publication practices. For this
reason, the generated teams can be called “theoretically possible teams” and comparable
with actual teams in terms of their characteristics.
4.3. Related Works
Notice that the growth of collaboration networks by the team formation
mechanism proposed in the previous section resembles the growth of the Web by the
vertex copying process (Kleinberg et al., 1999). In the vertex copying model, a new node
arrives at each point in time, selects an existing node uniformly at random, and copies the
linkage pattern of the chosen node with or without mutation (see Section 2.3.4). The
pattern of the chosen node’s outgoing links is represented by a column of the adjacency
matrix, and copying the pattern can be seen as adding to the matrix a new column that is
a copy of the chosen column. In the current model bipartite collaboration networks grow
in a similar manner. An authoring team is represented by a column of a bipartite matrix,
and repeated collaboration of the team means copying the column and adding to the
matrix as a new column. When the replication fails, the members of an existing team are
replaced with other authors, resulting in the growth and structural change of collaboration
networks.
110
The column copying process generates power-law distributions of the number of
papers per author 𝑞 𝑖 (Lotka, 1926; Simon, 1955) and of the collaborative connections per
author 𝑘 𝑖 (Barabási et al., 2002; Newman, 2001b), both of which are empirically
observed and predicted in previous studies. In the current model, each column of a
bipartite network is chosen for replication uniformly at random (Step 1), which represents
a team of authors who worked together to publish a paper. Therefore, the probability of a
particular author being a member of the chosen team and writing a new paper is
proportional to the number of the author’s published papers. In short, authors with more
publications are more likely to write new papers as in the preferential attachment process.
Formally, the growth rate of the number of papers of author i at time t is equal to the
proportion of the number of i’s papers over the total sum of all authors’ papers:
𝑑 𝑞 𝑖 𝑑𝑡 =
𝑞 𝑖 ∑𝑞 𝑗 𝑗 ,
where 𝑞 𝑖 is the number of papers of author i. If a non-zero positive mutation rate is
assumed (𝑚 ̅ > 0) as in Step 2 of the current model, a new author will join and write a
new paper every 1/𝑚 ̅ time. Then, as 𝑡 → ∞, the (relative) frequency of authors with q
published papers will follow a power law distribution:
𝑃 (𝑞 )~𝑞 −(
1
1−𝑚 ̅
+1)
,
as proved by Simon (1955).
The team formation mechanism also leads to power-law distributions of the
collaborative connections per author (Barabási et al., 2002; Newman, 2001b; see also
Barabási & Albert, 1999). The current model assumes that a one-mode network of
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authors grows by the addition of a new author every 𝑚 ̅ time, as described in Barabási and
Albert’s (1999) scale-free network model. Further, such a new author is assumed to work
together and build new collaborative links with existing authors (the green lines in Figure
4.2c). In that case, authors with more papers and more collaborative links are more likely
than those with fewer papers to be selected as members of repeating teams, which the
new author will join. Therefore, new collaborative links from new authors are more likely
to be formed to existing authors with more collaborative links. More precisely, the
probability that new authors will form collaborative links to existing authors is
proportional to the degree of the existing authors in the collaboration network. This is
consistent with the linear preferential attachment assumption in the emergence of scale-
free networks. Formally, the growth rate of connectivity of author i at t is equal to the
proportion of i’s connectivity of the total sum of all authors’ connectivity:
𝑑 𝑘 𝑖 𝑑𝑡 =
𝑘 𝑖 ∑𝑘 𝑗 𝑗 ,
where 𝑘 𝑖 is the number of authors who worked with author i at least one time. If this
preferential attachment is combined with a linear growth of nodes (i.e., one new node
arrives at every 1/𝑚 ̅ time), the (relative) frequency of authors with k collaborative links
will follow a power law distribution:
𝑃 (𝑘 )~𝑘 −(
1
𝑚 ̅
+1)
.
The team formation mechanism can capture the relational inertia in collaboration
networks (Ramasco & Morris, 2006). Because a team (i.e., a column of matrix B), rather
than an author, is the unit of selection for replication in the model, a group of authors
112
who have worked together for many times is more likely to be selected together to write a
new paper. In other words, authors tend to work with their previous team members for a
new project rather than with strangers (Guimerà et al., 2005). In addition, as described in
Step 4, the relational strength of dyads (𝑤 𝑖𝑗
) further facilitates repeated collaborations
between those who have prior collaboration experiences. When a high level of relational
inertia is expected, the collaboration networks will grow to be highly clustered,
reproducing the high levels of transitivity observed in empirical collaboration networks
(Newman, 2001a)
Thus far, it has been shown that the mechanism of team formation is sufficient to
generate power-law distributions of the number of papers per author 𝑃 (𝑞 ) and of the
collaborative connections per author 𝑃 (𝑘 ). However, empirical evidence suggests that
these two kinds of distributions are not exactly power-law. Instead, empirical data are
better fitted by power-law distributions with exponential cut-offs (Newman, 2001c).
More specifically, for the number of papers per author,
𝑃 (𝑞 )~𝑞 −(
1
1−𝑚 ̅
+1)
𝑒 𝑞 /𝑞 ∗
and for the collaborative connections per author
𝑃 (𝑘 )~𝑘 −(
1
𝑚 ̅
+1)
𝑒 𝑘 /𝑘 ∗
,
where 𝑞 ∗
and 𝑘 ∗
are the cutoff values of q and k, respectively, due to some externally
imposed constraints on the growth mechanism.
One possible explanation of these cutoff values in empirical collaboration
networks is that it arises as a result of the finite professional lifetime of individual
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authors. In other words, every author will eventually retire at some point and will stop
collaborating and publishing papers. If the finite professional lifetime is taken into
account, one main consequence is immediately expected: not all authors may work
together, but only those who are contemporaneous. This in turn limits the maximal
numbers of papers and collaborative connections that authors can have for their entire
professional lifetimes, namely, the “nodal maximal carrying capacity” of collaboration
networks (Monge et al., 2008).
Ramasco and Morris (2006) proposed to estimate professional lifetime using the
techniques used in survival analysis. However, their estimation method may not be
applicable to empirical data due to the censoring problems inherent in most longitudinal
data (Klein, Houwelingen, Ibrahim, & Scheike, 2013). The publication year of an
author’s first paper can be seen as the “birth year” of the author, and that of the last paper
as the “death year” of the author. Further, the duration between the birth and death years
can be seen as the author’s lifetime. However, no author can publish a paper in a journal
before the journal starts to publish. If it is the case, the estimation of birth year is
necessarily inaccurate. Such a problem is called a “left-censoring” problem. On the other
hand, the publication year of an author’s last paper in a given data set does not
necessarily indicate the death year of the author. The author might publish another paper
in the following years. Thus, the estimation of death year may not be accurate, either.
Such a problem is called a “right-censoring” problem. Due to the double censoring
problem of empirical data, Ramasco and Morris’s (2006) method does not produce a
reliable survival function. A solution for the censoring problem will be discussed in detail
in Section 5.3.
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4.4. The Decomposition of the Structural Change of Collaboration Networks
The team formation mechanism generates possible authoring teams that are
formed by the replication of existing teams with mutation, each of whose addition to an
existing collaboration network results in a unique collaboration network. All the resultant
networks represent ones into which the existing network would possibly change in the
very next point in time (Figure 4.3). The possible networks would be fairly similar to one
another because they all stem from a common ancestral network (one at the top of Figure
4.3). Nevertheless, they would not be completely identical to one another because slightly
different authoring teams were added as the result of the mutation process.
In the complete absence of selection pressure, that is, when every paper is equally
likely to be selected for publication, every possible authoring team is equally likely to be
added to the existing network regardless of their abilities to produce acceptable papers.
Accordingly, every possible network is equally likely to be realized as well. Suppose that
an authoring team is selected uniformly at random. In that case, the most “typical” among
all the possible networks is expected to be realized. Further, the structural property z of
the most typical network can be characterized by the median
12
of z of all the possible
networks, denoted by 𝑧 ̂ 𝑡 (the blue dashed vertical bars in Figure 4.3). In that sense 𝑧 ̂ 𝑡 can
be understood as the expected structure of collaboration networks when they grow in the
absence of selection pressure. The expected structure of collaboration networks is likely
to be different from that of their ancestral networks (i.e., 𝑧 ̂ 𝑡 ≠ 𝑧 𝑡 −1
), unless all the
12
It is known that the mean is a reliable summary of a given distribution. By the central limit theorem the
mean of a sample from the distribution is an unbiased estimator of the distribution when the sample size is
large enough (typically, n > 30). In the current context, however, one and only one network is sampled
from the alternative networks at a given point in time (i.e., n = 1). In that case, the median is better to
characterize the distribution, because the sampled network will be larger or smaller than the median with an
equal probability. See Lee (1995) for more technical details.
115
authoring teams are generated by the exact replication of existing teams. Such differences
can be interpreted as the structural change of collaboration networks by the growth of the
networks, or more precisely, the change due to the replication and mutation processes
(𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
).
Figure 4.3. The decomposition of the structural change of collaboration networks into (1)
the change due to the replication and mutation processes and (2) the change due to the
selection mechanism.
On the other hand, when the range of acceptable papers is narrow (i.e., under
strong selection), only a few authoring teams can produce acceptable papers and thus join
116
the existing network (the red bubble in Figure 4.3) as discussed in Chapter 3.
Accordingly, only a small subset of possible networks are expected to be realized (e.g.,
the network in the red box in Figure 4.3). The networks expected under strong selection
will be different from the ancestral network because new authoring teams are added and
different from those that grow in the absence of selection pressure because authoring
teams are selected from a more specific set of “capable” authoring teams rather than any
possible authoring teams. More specifically, the networks under strong selection are
expected to show higher values of the largest eigenvalue and modularity and smaller
spectral gap than those expected in the complete absence of selection pressure as
suggested by the simulation results in Chapter 3.
From this perspective the observed structural change of a collaboration network,
that is, the difference in structural property z observed at two different points in time
(∆𝑧 = 𝑧 𝑡 − 𝑧 𝑡 −1
: the green upward arrow in Figure 4.4), can be decomposed into two
parts (Figure 4.4). The first part is the change due to the replication and mutation
processes in the team formation mechanism (i.e., 𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
: the blue upward arrow in
Figure 4.4). When the replication and mutation processes do not fully account for the
observed structural change (i.e., 𝑧 ̂ 𝑡 ≠ 𝑧 𝑡 ), the remaining amount of the change could be
explained by the selection mechanism (i.e., 𝑧 𝑡 − 𝑧 ̂ 𝑡 : the red upward arrow in Figure 4.4).
The sum of these two parts are always equal to the total amount of the observed change:
∆𝑧 = 𝑧 𝑡 − 𝑧 𝑡 −1
= (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
)+ (𝑧 𝑡 − 𝑧 ̂ 𝑡 ). (4.1)
117
Figure 4.4. The decomposition of the observed change in structural property z by the
source of change.
Notice that 𝑧 𝑡 and 𝑧 𝑡 −1
denote the structural property z of a network observed at
two different points in time, both of which are always empirically measureable as long as
longitudinal network data are available. The difference between the two values (∆𝑧 =
𝑧 𝑡 − 𝑧 𝑡 −1
) is divided by the quantity 𝑧 ̂ 𝑡 that measures the expected structure of a
collaboration network in the absence of selection pressures. In a sense, the second term in
Equation 4.1 can be seen as the amount of the observed change unexplained by the team
formation mechanism or simply the residuals of the current model in the jargon of
statistical analysis. In conventional statistical analysis, the residuals of a statistical model
are assumed to be randomly distributed around zero (Bollen, 1989). If the expected value
of the residuals is not equal to zero, the prediction made by the model is inaccurate.
118
When the residuals of a regression model have a non-zero expected value, for example,
the model can be improved just by adding a constant term, often called the y-intercept.
Because the purpose of a regression model is to understand the relationships among
variables, and because the constant term is not correlated with the dependent variable or
explanatory variables, it is usually considered to have no significant meaning and
excluded from the analysis. However, it is worth noting that the constant term of a
regression model, or any statistical models in general, represents the common and
consistent bias inherent in all the observations, which cannot be explained by any of
explanatory variables. For this reason, the constant term is essential to predict the
observations based on the regression model, whereas it is unimportant to examine the
relationships among variables.
Technically, the effects of the selection mechanism on the structural change of
collaboration networks is defined as the residuals of the current growth model (i.e., 𝑧 𝑡 −
𝑧 ̂ 𝑡 ). However, the present study expects that the residuals are not randomly distributed
around zero (i.e., 0 = 𝑧 𝑡 − 𝑧 ̂ 𝑡 , Figure 4.5a) but instead exhibit nonrandom patterns
(Figure 4.5b). More specifically, the residuals are expected to be consistently biased
toward high levels of the largest eigenvalue and modularity but low levels of spectral gap
due to the selection mechanism. The following chapter will examine empirical
collaboration networks to confirm this expectation.
119
(a) Random residuals
(b) Biased residuals
Figure 4.4. The patterns in the residuals of the growth model: (a) random and (b) biased.
120
CHAPTER 5: THE EVOUTION OF EMPIRICAL SCIENTIFIC
COLLABORATION NETWORKS
The models of collaboration networks proposed in the previous chapters
incorporate the three fundamental mechanisms of evolutionary dynamics—replication,
mutation, and selection. As a basic analytical unit of social and communication relations
among researchers, new authoring teams are formed by replicating the collaborative
connections among the members of existing teams. However, new teams are not always
identical to the existing teams in terms of membership structure. Mutation occurs when
those leaving the teams are replaced with new members. Because the addition of different
authoring teams results in different collaboration networks, the team formation
mechanism by replication and mutation generates an ensemble of possible networks into
which an observed network would have changed. Not all authoring teams would succeed
in publishing papers. Instead, teams who can produce acceptable papers can join an
existing collaboration network and eventually alter the structural properties of the
network. As a result of the operation of the selection mechanism, collaboration networks
are expected to exhibit higher values of the largest eigenvalue and modularity and smaller
spectral gap under strong selection than under weak or no selection.
Based on the proposed model, this chapter examines the evolutionary change of
empirical collaboration networks. Eight journals across disciplines were chosen for
analysis: (1) the Journal of Communication (2) Philosophy of Science, (3) Chemical
Reviews, (4) Trends in Neurosciences, (5) Biometrika, (6) the Journal of Theoretical
Biology, (7) the American Journal of Sociology, and (8) Science. Table 5.1 summarizes
the basic information of the journals.
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Table 5.1. The eight journals sampled for analysis.
Journal (abbreviation) Discipline
Year of the
first issue
No.
papers
No.
authors
Journal of Communication
(JOC)
Social Science 1951 2,802 3,098
Philosophy of Science
(PHILO)
Arts &
Humanities
1934 4,532 2,502
Chemical Reviews
(CHEM)
Chemistry 1924 4,569 8,113
Trends in Neurosciences
(NEURO)
Medicine 1978 5,013 6,537
Biometrika (BIOMET) Mathematics 1901 7,196 5,110
Journal of Theoretical
Biology (JTB)
Biology 1961 12,121 16,255
American Journal of
Sociology (AJS)
Social Science 1895 21,594 9,790
Science (SCI) Multidisciplinary 1880 110,730 110,392
Total 168,557 161,797
5.1. Data Collection
For each of the eight journals, the bibliographic information of all the papers that
were published between the year of the first issue and 2010 was collected from the
database of Microsoft Academic Search (MAS, http://academic.research.microsoft.com).
The MAS is a free public search engine for academic papers and literature, developed by
Microsoft Research. The database includes over 39.9 million indexed publications and
19.9 million authors. Although largely functional, the service has not been updated since
2013 and seen a marked decline in the number of indexed documents since 2011. For this
reason, the current data set included papers that had been published before 2011. Despite
122
several limitations of the MAS database, one of its strengths is that it resolves the
problems with identifying distinctive authors, which has been a common problem of
previous research on collaboration networks (see the discussion in Newman, 2001c).
It is not uncommon that a single author reports his or her name differently on
different papers. For example, Peter Monge the communication scholar reported his name
in three different ways even within a single journal: Peter Monge, Peter R. Monge, and P.
Monge. These three different names should be considered as the same person. Otherwise,
the number of nodes in the collaboration network will be overestimated and other
network measures may be misestimated. Another problem is when there are two or more
authors who have the same name. For example, there is another Peter R. Monge who is a
pharmacologist and has published no single paper in communication journals or any other
social scientific journals. The pharmacologist should be distinguished from the
communication scholar. Otherwise, the number of nodes in the collaboration network
will be underestimated and other network measures may be misestimated. The failure to
identify distinctive authors can cause problems in building collaboration networks from
bibliographic information. Because every author is uniquely indexed in the MAS
database, such problems can be handled easily.
Another strength of the MAS database is that it provides a set of APIs (application
program interfaces) that make it possible to collect all the papers that individual authors
have published across 21,989 journals and conference proceedings. This function was
particularly important to measure the professional lifetime of authors, because it allows
us to specify when authors published their very first and last papers and how many papers
between them. Section 5.3 discusses the measurement strategy in detail.
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Among the collected papers “uncitable items” were excluded, such as editorials,
abstracts in foreign languages, calls for papers, author guidelines, obituaries, etc.
(Moustafa, 2015). The final data set included 168,557 papers and 161,797 distinctive
authors. Further, all the papers written by those authors were collected (“citable” items
only). The total number of such papers was 4,917,088 (on average 30.39 papers per
author).
5.2. The Growth of Empirical Collaboration Networks
Based on the bibliographic information, eight longitudinal collaboration networks
were constructed. Figure 5.1 presents the growth of nodes and collaboration links of the
networks over time. Note that the y-axes of the plots are logarithmic, and thus, a straight
line indicates exponential growth. In terms of population growth, exponential growth is a
result of simple replication without any external constraints (e.g., limited supplies of
resources: see Section 2.2.1). On the other hand, the deviation from a straight line,
whether sub- or super-linear, implies the presence of external constraints imposed on the
replication process. In most cases, nodes and collaboration links grew sub-linearly. This
may imply that nodes could not freely enter the networks, neither could edges be formed
between any pairs of nodes, suggesting that growth by replication was inhibited by some
external constraints. On the other hand, the collaboration links of Science (the black solid
line), Biometrika (the blue solid lines) and Trends in Neurosciences (the green solid line
in Figure 5.1b) grew super-linearly, at least during some periods of time. This suggests
that there were some underlying mechanisms that facilitated collaborations among
authors in those journals.
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(a)
(b)
Figure 5.1. The growth of (a) nodes and (b) collaborative connections over time on the
logarithmic scale of the y-axis.
Next, the changes in the transitivity of the eight networks were examined. All the
networks became more transitive and clustered over time (Figure 5.2a). This implies the
subgroups of authors, perhaps, subareas of the journals emerged over time. However, the
transitivity of the networks increased at different rates. Biometrika (the blue solid line),
the American Journal of Sociology (the red solid line), and Philosophy of Science (the
black dashed line) became clustered more slowly than other journals. On the other hand,
Trends in Neurosciences (the green solid line) and Chemical Reviews (the green dashed
line) became clustered at the fastest rates.
125
(a)
(b)
Figure 5.2. The changes in (a) the transitivity and (b) the relative size of the largest
connected component over time.
In addition, the changes in the relative size of the largest connected components
of the networks were examined. The existence of a large connected component in a
network indicates that the network is well connected as a whole and that a large
proportion of nodes are reachable from each other (Newman, 2010). Further, it can be
interpreted as the emergence of “invisible colleges” in the context of scientific
collaboration networks (D., J de Solla Price & Beaver, 1966). A U-shaped pattern was
commonly found in the changes in the size of largest connected components across all the
eight networks (Figure 5.2b). The size of the largest components sharply decreased
during the early stages of the development (i.e., the disintegration of the networks) but
increased later on (i.e., the reintegration of the networks). Both the American Journal of
Sociology (the red solid line) and Philosophy of Science (the black dashed line) remained
126
relatively fragmented, however. This is in large part because of the dominance of single-
authored papers in those journals (92.5% and 94.3% respectively; the overall average was
69.5% in all the other journals).
The simple descriptive analysis of the collaboration networks provides interesting
insights. First, the non-linear growth patterns of nodes and edges (Figure 5.1) suggest that
both nodes and edges did not grow by simple replication processes. Second, the
increasing transitivity of the collaboration networks over time (Figure 5.2a) implies that
the growth of the networks were not pure randomness, because random growth
mechanisms, such as Erdős and Rényi’s (1960) model and Barabási and Albert’s (1999)
model, cannot generate high levels of transitivity. Third, the changes in the size of the
largest components in the U-shaped pattern (Figure 5.2b) suggests that the collaboration
networks had been (re)integrated by some self-organizing mechanism.
5.3. Generating Authoring Teams and Possible Networks
In order to investigate the evolutionary mechanism that operated on the eight
collaboration networks, possible authoring teams and networks were generated by the
team formation mechanism developed in Chapter 4. The mechanism requires six
parameters (Table 4.1), all of which can be obtained from empirical data. The first
parameter is the number of authors per paper or team size 𝑃 𝑛 (𝑛 ) (Figure 5.3a). The
second parameter is the number of papers published by authors in the journals or
publication experience 𝑃 𝑛 (𝑛 ) (Figure 5.3b). The third parameter is the number of
collaborative connections or weighted degree distribution 𝑃 𝑤 (𝑤 ) (Figure 5.3c). The forth
parameter is the number of distinctive collaborators or unweighted degree distribution
𝑃 𝑘 (𝑘 ) (Figure 5.3d).
127
(a) Team size
(b) Publication experience
(c) Weighted degree distribution
(d) Unweighted degree distirbution
Figure 5.3. The distributions of (a) team size and (b) publication experience and (c) the
weighted and (d) unweighted degree distributions of the eight collaboration networks on
double logarithmic scales.
All distributions plotted on double logarithmic scales were better fitted by power-
law distributions with exponential cut-offs (Newman, 2001c) rather than straight lines,
which again implies some external constraints imposed on the growth mechanism. As
discussed in Section 4.3, one possible explanation for the exponential cutoffs is the finite
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professional lifetime of individual authors. Every author can work with a limited number
of collaborators for a finite number of times, and thus, produce a finite number of papers
for their lifetimes. To incorporate this idea into the model, authors’ professional lifetimes
were measured as follows. First, all the papers published in any venues by an author were
collected. Second, the time period between the first and last papers were measured in
years and divided by the number of all the papers. This gave the average time interval
between papers, which can be interpreted to indicate the average amount of time required
for the author to prepare a new paper. The professional lifetime of the author was defined
as the duration between the first and last papers plus and minus the average interval.
Figure 5.4 illustrates an example of measuring an author’s professional lifetime.
The author published his/her first paper in 1951 and last paper in 1974 and produced 10
other papers in between. On average, it took two years (= 24 years / 12 papers) for the
author to produce a new paper. So, it is reasonable to expect that it took two years for the
author to prepare the first paper. Thus, even before the first paper was published, the
author had been active for two years so that the author would have been able to join
authoring teams. Similarly, the author would be active for another two years even after
the last paper were published. But the author failed to publish any additional papers.
Thus, the author can be considered as to be active between 1949 and 1976 (for 28 years).
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Figure 5.4. Measuring the professional lifetime of an author.
The active authors in each year were identified and used to generate new
authoring teams. Figure 5.5 presents the number of active authors over time.
Figure 5.5. The number of active authors over time.
The last two parameters—relational inertia R and the average proportion of new
authors in teams 𝑚 ̅—are presented in Figure 5.6. Interestingly, these two parameters
were negatively correlated, although the coefficient was not statistically significant
(Pearson’s r = –.65; t = –2 .10, df = 6, p = .08). It appeared to be unlikely that authors
130
with no previous publication experience (i.e., new authors) published new papers in the
American Journal of Sociology and Philosophy of Science (the two red dots on the top-
left), whereas collaborations were frequently repeated in those journals. This implies that
mutations were rare in those journals and the structure of the collaboration networks
changed very slowly over time, as shown in Figure 5.2. In Chemical Reviews, Trends in
Neurosciences, and the Journal of Theoretical Biology (the three blue dots on the bottom-
right), on the other hand, repeated collaborations were relatively rare, and new authors
entered frequently. As a result, the structure of those networks changed very rapidly
(Figure 5.2)
Figure 5.6. The average rate of new authors and the relational inertia of the eight
collaboration networks.
Using the six parameters obtained from the empirical data, authoring teams were
generated by the team formation mechanism as follows:
131
1. An existing authoring team was selected uniformly at random and inactive
members of the team, if any, were removed.
2. Of the remaining members l members were selected with probability proportional
to their tendencies of repeated collaboration 𝑟 𝑖 and continued to work in the team.
The (n – l) empty spots of the team were filled in two different ways.
3. A value of m was taken from the conditional probability distribution 𝑃 𝑚 |𝑛 (𝑚 |𝑛 ).
m authors with no previous publication experience were randomly sampled from
the pool of the active authors in the given year.
4. The remaining (n – l – m) spots were filled with experienced authors in the author
pool in the given year, depending on their prior experiences of collaboration with
the l continuing authors 𝑤 𝑖𝑗
. When none of the continuing authors had previous
collaborators, experienced authors were selected according to their publication
experience 𝑞 𝑖 .
By adding this newly formed team to the existing network, a possible alternative network
is generated.
For each journal, the number of actual authoring teams in a given year was
counted and an equal number of authoring teams were generated. By adding those
generated teams to the network observed during the previous year, a possible network
was constructed. This procedure was iterated 1,000 times to generate a total of 1,000
possible networks in a given year. Note that because a reasonable number of existing
teams are required to carry out the first step of the team formation (i.e., sampling for
replication), the simulation routine was started from the second year of each journal. The
132
simulation routine for team generation was implemented by using R (ver. 3.1.2) the
programming language (R Core Team, 2014). All scripts are available upon request.
5.3. The Structural Change of Empirical Collaboration Networks
5.3.1. The Analysis of Observed Change
The structural properties of the collaboration networks were quantified by three
different measures—(1) the largest eigenvalue 𝜆 1
, (2) spectral gap Δ𝜆 (= 𝜆 1
− 𝜆 2
,
difference between the largest and the second largest eigenvalues of the adjacency matrix
of a network), and (3) the modularity of the largest connected component of a network.
The arpack function built in the R-igraph package (Csardi & Nepusz, 2006) was used
to compute the eigenvalues of the adjacency matrices of the observed and alternative
networks. According to the authors of the function, it is designed to compute a few
eigenvalues and corresponding eigenvectors of a general square matrix. It is most
appropriate for large sparse matrices. Modularity was computed by the
fastgreedy.community function based on the optimization algorithm proposed by
Clauset et al. (2004; for technical details see Csardi & Nepusz, 2006).
For each journal, the observed change in structural properties was decomposed
into two parts—(1) the change due to the replication and mutation processes and (2) the
change due to the selection mechanism. The observed change was simply measured by
the difference of the network connectedness in a given year from that in the previous year
(i.e., ∆𝑧 = 𝑧 𝑡 − 𝑧 𝑡 −1
) and decomposed into two parts:
∆𝑧 = (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
)+ (𝑧 𝑡 − 𝑧 ̂ 𝑡 ),
133
where 𝑧 ̂ 𝑡 is the expected structural properties and quantified by the medians of the
structural properties of all the 1,000 possible networks generated by the team formation
mechanism in a given year. The first part (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
) denotes the amount of the change
due to the replication and mutation. The second part (𝑧 𝑡 − 𝑧 ̂ 𝑡 ) is the amount of the
change due to selection.
5.3.2. The Largest Eigenvalue
The largest eigenvalue 𝜆 1
of the adjacency matrix of an undirected network (as in
collaboration networks) measures its robustness against random failures of nodes or
edges, that is, the degree to which the network remains connected after a fraction of
nodes or edges are deleted (Section 3.4.2). The simulation results in Chapter 3 suggested
that collaboration networks with a large value of 𝜆 1
are likely to occur under strong
selection. For each journal the change in the largest eigenvalue was computed in every
year and decomposed into the two parts, ∆𝑧 = (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
)+ (𝑧 𝑡 − 𝑧 ̂ 𝑡 ). The results are
summarized in Figures 5.7 to 5.14.
In each figure, the black curve indicates the largest eigenvalue of the actual
collaboration network. The largest eigenvalues of all the eight collaboration networks
increased over time but at different rates. The largest eigenvalue of the collaboration
network of Science increased at the fastest rate (90.3 per year). This is in large part
because the journal is the oldest and has published the largest number of papers but not
necessarily because the journal had the most restrictive selection criteria or imposed the
strongest selection pressure. The median of the largest eigenvalues of the 1,000 possible
networks in a given year is presented as a blue point in each figure, representing the
expected largest eigenvalue in the absence of selection pressures. The 95% confidence
134
intervals are presented as the blue dashed curves. Across all the journals the observed
largest eigenvalues were close to the expected largest eigenvalues and fell within the 95
confidence intervals.
Nevertheless, slight but consistent deviations were found between the observed
and expected largest eigenvalues across all the journals. Specifically, the black curves
were almost always located above the blue curves across all the journals. To further
investigate the deviations, the percentile ranks of the observed eigenvalues were
measured. In general, the percentile rank of a score is defined as the proportion of scores
in its frequency distribution that are equal to or lower than the score, telling exactly
where the score is located relative to all the other scores in the distribution (Devore &
Berk, 2007). A common example of the use of percentile rank is the z-transformation
(also called “standardization”) that converts raw scores into z-scores by dividing the
standard deviation of the raw scores so that the relative distances of the raw scores from
the mean can be assessed regardless of the distributions of the raw scores. The percentile
rank of the largest eigenvalue of an observed network can be defined as the proportion of
possible networks whose largest eigenvalues were equal to or less than it. The percentile
rank of the observed network is equal to .50 (i.e., the median), when the network is as
well connected as a “typical” network among all the possible networks. The percentile
rank is greater than .50 when the observed network is better connected than a majority of
the possible networks.
In two journals, Chemical Reviews (Figure 5.9) and the Journal of Theoretical
Biology (Figure 5.12), the black curves are clearly above the blue curves. Most of the
percentile ranks of the observed largest eigenvalues were above .50 with a few exceptions
135
(7.0% and 4.1%, respectively). In Chemical Reviews, the total amount of the observed
change in the largest eigenvalue was 1,901.85 between 1925 and 2010. Of it 58.4% was
due to the replication and mutation processes, whereas 41.6% was due to the selection
mechanism (Figure 5.9). In the Journal of Theoretical Biology, the total amount of the
observed change was 1,303.65 between 1962 and 2010, 59.9% of which was due to the
replication and mutation processes, whereas 40.1% was due to the selection mechanism
(Figure 5.12). In two other journals, Philosophy of Science (Figure 5.8) and the American
Journal of Sociology (Figure 5.13), the black and blue curves are almost overlapping.
This means that the observed change in the largest eigenvalue was mainly due to the
replication and mutation processes (83.0% and 83.8%, respectively) rather than the
selection mechanisms (17.0% and 16.2%, respectively). Table 5.2 summarizes the
changes in the largest eigenvalues of the eight collaboration networks.
(a)
(b)
Total change:
188.67 (100.0%)
By
replication/mutation:
144.27 (76.5%)
By selection:
44.40 (23.5%)
Figure 5.7. (a) The change in the largest eigenvalue λ1 of the collaboration network of the
Journal of Communication between 1952 and 2010 and (b) the percentile ranks of the
observed values of λ1.
136
(a)
(b)
Total change:
93.33 (100.0%)
By
replication/mutation:
77.47 (83.0%)
By selection:
15.85 (17.0%)
Figure 5.8. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Philosophy of Science between 1935 and 2010 and (b) the percentile ranks of
the observed values of λ1.
(a)
(b)
Total change:
1901.85 (100.0%)
By
replication/mutation:
1111.21 (58.4%)
By selection:
790.63 (41.6%)
Figure 5.9. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Chemical Reviews between 1925 and 2010 and (b) the percentile ranks of the
observed values of λ1.
137
(a)
(b)
Total change:
991.91 (100.0%)
By
replication/mutation:
600.89 (60.6%)
By selection:
391.01 (39.4%)
Figure 5.10. The observed change in the largest eigenvalue λ1 of the collaboration
network of Trends in Neurosciences between 1979 and 2010 and (b) the percentile ranks
of the observed values of λ1.
(a)
(b)
Total change:
397.18 (100.0%)
By
replication/mutation:
274.89 (69.2%)
By selection:
122.29 (30.8%)
Figure 5.11. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of Biometrika between 1902 and 2010 and (b) the percentile ranks of the
observed values of λ1.
138
(a)
(b)
Total change:
1303.65 (100.0%)
By replication with
mutation:
780.33 (59.9%)
By selection:
532.3248 (40.1%)
Figure 5.12. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of the Journal of Theoretical Biology between 1962 and 2010 and (b) the
percentile ranks of the observed values of λ1.
(a)
(b)
Total change:
169.08 (100.0%)
By
replication/mutation:
141.61 (83.8%)
By selection:
27.47 (16.2%)
Figure 5.13. (a) The observed change in the largest eigenvalue λ1 of the collaboration
network of the American Journal of Sociology between 1896 and 2010 and (b) the
percentile ranks of the observed values of λ1.
139
(a)
(b)
Total change:
11720.57 (100.0%)
By replication with
mutation:
8307.73 (70.9%)
By selection:
3412.838 (29.1%)
Figure 5.14. The observed change in the largest eigenvalue λ1 of the collaboration
network of Science between 1881 and 2010 and (b) the percentile ranks of the observed
values of λ1.
Table 5.2. The changes in the largest eigenvalue λ1 of the collaboration networks of the
eight journals.
Journals
Change in the largest eigenvalue 𝝀 𝟏
Total (%)
By replication and
mutation (%)
By selection (%)
Journal of Communication 188.67 (100.0%) 144.27 (76.5%) 44.40 (23.5%)
Philosophy of Science 93.33 (100.0%) 77.47 (83.0%) 15.85 (17.0%)
Chemical Reviews 1,901.85 (100.0%) 1,111.21 (58.4%) 790.64 (41.6%)
Trends in Neurosciences 991.91 (100.0%) 600.89 (60.6%) 391.01 (39.4%)
Biometrika 397.18 (100.0%) 274.89 (69.2%) 122.29 (30.8%)
Journal of Theoretical Biology 1,303.65 (100.0%) 780.33 (59.9%) 532.32 (40.1%)
American Journal of Sociology 169.08 (100.0%) 141.61 (83.8%) 27.47 (16.2%)
Science 11,720.75 (100.0%) 8,307.73 (70.9%) 3412.84 (29.1%)
140
5.3.3. Spectral gap
The spectral gap Δ𝜆 , defined as the difference between the first and second largest
eigenvalues of the adjacency matrix of a network, measures network robustness against
targeted attacks (see Section 3.4.2). The spectral gap of a network is maximal when every
node is connected to each other (i.e., a complete network). Networks characterized by
high levels of spectral gap are highly robust but inefficient in terms of connection costs.
For this reason, it is expected that networks evolve into ones with low levels of spectral
gap under strong selection, as shown in the simulation results in Section 3.4.2. The
changes in the spectral gap of the eight collaboration networks were measured and
decomposed into two parts in the same way as done in the previous section. The results
are summarized in Figures 5.15 to 5.21.
The results suggest that the spectral gap of all the collaboration networks had
substantially increased over time. For example, the spectral gap of the collaboration
network of Science increased to 1,774.28 in 2010, the largest among the eight networks.
However, the spectral gap of a network could be increased largely due to the replication
and mutation processes, even when the selection mechanism operates to lower the level
of spectral gap. The decomposition of the observed change confirmed this expectation.
Across all the collaboration networks, the replication and mutation processes
substantially increased the spectral gap of the networks. However, the selection
mechanism operated in the opposite direction, suppressing the increased spectral gap. For
example, the spectral gap of the collaboration network of Trends in Neurosciences had
increased by 340.43 due to the replication and mutation processes between 1979 and
2010 (Figure 5.18). However, those with low levels of spectral gap were repeatedly
141
observed among the possible networks. As a result, the spectral gap of the actual network
did not increase as much as it would have increased in the absence of selection pressure.
Specifically, 41.0% of the increased spectral gap by the replication and mutation
processes was reduced by the selection mechanism.
The suppressing effects of the selection mechanism were observed across all the
journals. In the four journals, Chemical Reviews (Figure 5.17), Trends in Neurosciences
(Figure 5.18), the Journal of Theoretical Biology (Figure 5.20), and Science (Figure
5.22), more than 30% of the increased spectral gap was suppressed by the selection
mechanism. On the other hand, the suppressing effects were smallest in the collaboration
network of Philosophy of Science (Figure 5.16). Of the increased spectral gap by the
replication and mutation processes, only 11.8% were reduced by the selection
mechanism. Table 5.3 summarizes the changes in the spectral gap of the eight
collaboration networks.
142
(a)
(b)
Total change:
93.75 (81.0%)
By
replication/mutation:
115.79 (100.0%)
By selection:
-22.04 (-19.0%)
Figure 5.15. (a) The observed change in the spectral gap Δλ of the collaboration network
of the Journal of Communication between 1952 and 2010 and (b) the percentile ranks of
the observed values of Δλ.
(a)
(b)
Total change:
56.11 (88.1%)
By
replication/mutation:
63.64 (100.0%)
By selection:
-7.53 (-11.8%)
Figure 5.16. (a) The observed change in the spectral gap Δλ of the collaboration network
of Philosophy of Science between 1935 and 2010 and (b) the percentile ranks of the
observed values of Δλ.
143
(a)
(b)
Total change:
188.92 (63.5%)
By
replication/mutation:
297.30 (100.0%)
By selection:
-108.38 (-36.5%)
Figure 5.17. (a) The observed change in the spectral gap Δλ of the collaboration network
of Chemical Reviews between 1925 and 2010 and (b) the percentile ranks of the observed
values of Δλ.
(a)
(b)
Total change:
200.83 (59.0%)
By
replication/mutation:
340.43 (100.0%)
By selection:
-139.59 (-41.0%)
Figure 5.18. (a) The observed change in the spectral gap Δλ of the collaboration network
of Trends in Neurosciences between 1979 and 2010 and (b) the percentile ranks of the
observed values of Δλ.
144
(a)
(b)
Total change:
185.24 (70.8%)
By
replication/mutation:
261.60 (100.0%)
By selection:
-76.36 (-29.2%)
Figure 5.19. (a) The observed change in the spectral gap Δλ of the collaboration network
of Biometrika between 1902 and 2010 and (b) the percentile ranks of the observed values
of Δλ.
(a)
(b)
Total change:
161.57 (60.3%)
By
replication/mutation:
267.89 (100.0%)
By selection:
-106.32 (-39.7%)
Figure 5.20. (a) The observed change in the spectral gap Δλ of the collaboration network
of the Journal of Theoretical Biology between 1962 and 2010 and (b) the percentile ranks
of the observed values of Δλ.
145
(a)
(b)
Total change:
94.36 (77.6%)
By
replication/mutation:
121.65 (100.0%)
By selection:
-27.29 (-22.43%)
Figure 5.21. (a) The observed change in the spectral gap Δλ of the collaboration network
of the American Journal of Sociology between 1986 and 2010 and (b) the percentile ranks
of the observed values of Δλ.
(a)
(b)
Total change:
1,774.28 (69.8%)
By
replication/mutation:
2,541.08 (100.0%)
By selection:
-776.80 (-30.2%)
Figure 5.22. (a) The observed change in the spectral gap Δλ of the collaboration network
of Science between 1881 and 2010 and (b) the percentile ranks of the observed values of
Δλ.
146
Table 5.3. The changes in the spectral gap Δλ of the collaboration networks of the eight
journals.
Journals
Change in spectral gap 𝚫 𝝀
Total (%)
By replication and
mutation (%)
By selection (%)
Journal of Communication 93.76 (81.0%) 115.79 (100.0%) –22.04 (–19.0%)
Philosophy of Science 56.11 (88.2%) 643.64 (100.0%) –7.53 (–11.8%)
Chemical Reviews 188.92 (63.5%) 297.30 (100.0%) –108.38 (–36.5%)
Trends in Neurosciences 200.84 (59.0%) 340.43 (100.0%) –139.59 (–41.0%)
Biometrika 185.24 (70.8%) 261.60 (100.0%) –76.36 (–29.2%)
Journal of Theoretical Biology 161.57 (60.3%) 267.98 (100.0%) –106.32 (–39.7%)
American Journal of Sociology 94.36 (77.6%) 121.65 (100.0%) –27.3 (–22.4%)
Science 1,774.28 (69.8%) 2,541.08 (100.0%) –766.80 (–30.2%)
5.3.4. Modularity
The third measure used for network connectedness was network modularity Q.
Modularity is measured as the fraction of the edges that fall within a given cluster minus
the expected such fraction if edges were distributed at random. In this dissertation, the
fastgreedy.community function in the R-igraph package (ver. 0.7.1) was used to
compute the modularity of the collaboration networks. The function detects network
partitioning that maximizes the modularity of a network and returns that modularity value
(Clauset et al., 2004; Csardi & Nepusz, 2006). The fast greedy modularity optimization
algorithm treats disconnected clusters or even isolated nodes as well-established modules.
Thus, highly fragmented networks tend to have high levels of modularity. For this reason,
147
only the largest connected components of the networks were considered to measure the
modularity of collaboration networks.
As summarized in Figures 5.23 to 5.30, in most cases, a considerable amount of
the change in network modularity was explained by the selection mechanism. However,
the replication and mutation processes also increased network modularity. For example,
the modularity of the collaboration network of the Journal of Communication increased
by .176 from .167 in 1952 to .346 in 2010 and 42.3% of the observed change was
explained by the replication and mutation processes (Figure 5.23). This suggests that
collaboration networks could have high modularity in the absence of selection pressures,
and further, network modularity was not necessarily the byproduct of the cost-efficiency
optimization. In fact, it is known from numerical experiments that even purely random
networks display intrinsic modularity and may be partitioned yielding high values of Q
(Guimerà, Sales-Pardo, & Amaral, 2004).
However, Reichardt and Bornholdt’s (2006) study suggests that random networks
can have high modularity when they are sparsely connected, and that the modularity of
random networks tend to diminish as the networks become denser. The change in the
modularity of the collaboration networks of Chemical Reviews (Figure 5.25) and Trends
in Neurosciences (Figure 5.26), which were the densest networks, demonstrates this
tendency. The replication and mutation processes in the two journals decreased network
modularity over time by 24.9% and 8.8%, respectively.
148
(a)
(b)
Total change:
.176 (100.0%)
By
replication/mutation:
.076 (42.3%)
By selection:
.102 (57.7%)
Figure 5.23. (a) The observed change in the modularity Q of the collaboration network of
the Journal of Communication between 1952 and 2010 and (b) the percentile ranks of the
observed values of Q.
(a)
(b)
Total change:
.167 (100.0%)
By
replication/mutation:
.074 (44.2%)
By selection:
.093 (55.8%)
Figure 5.24. (a) The observed change in the modularity Q of the collaboration network of
Philosophy of Science between 1935 and 2010 and (b) the percentile ranks of the
observed values of Q.
149
(a)
(b)
Total change:
.328 (100.0%)
By
replication/mutation:
-.082 (-25.0%)
By selection:
.410 (1.25%)
Figure 5.25. (a) The observed change in the modularity Q of the collaboration network of
Chemical Reviews between 1925 and 2010 and (b) the percentile ranks of the observed
values of Q.
(a)
(b)
Total change:
.362 (100.0%)
By
replication/mutation:
-.032 (-8.9%)
By selection:
.394 (108.9%)
Figure 5.26. (a) The observed change in the modularity Q of the collaboration network of
Trends in Neurosciences between 1979 and 2010 and (b) the percentile ranks of the
observed values of Q.
150
(a)
(b)
Total change:
.315 (100.0%)
By
replication/mutation:
.085 (26.8%)
By selection:
.231 (73.2%)
Figure 5.27. (a) The observed change in the modularity Q of the collaboration network of
Biometrika between 1902 and 2010 and (b) the percentile ranks of the observed values of
Q.
(a)
(b)
Total change:
.456 (100.0%)
By
replication/mutation:
.012 (2.7%)
By selection:
.444 (97.3%)
Figure 5.28. (a) The observed change in the modularity Q of the collaboration network of
the Journal of Theoretical Biology between 1962 and 2010 and (b) the percentile ranks of
the observed values of Q.
151
(a)
(b)
Total change:
.198 (100.0%)
By
replication/mutation:
.066 (33.2%)
By selection:
.132 (66.8%)
Figure 5.29. (a) The observed change in the modularity Q of the collaboration network of
the American Journal of Sociology between 1896 and 2010 and (b) the percentile ranks of
the observed values of Q.
(a)
(b)
Total change:
.298 (100.0%)
By
replication/mutation:
.000 (0.0%)
By selection:
.298 (100.0%)
Figure 5.30. (a) The observed change in the modularity Q of the collaboration network of
Science between 1881 and 2010 and (b) the percentile ranks of the observed values of Q.
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Table 5.4. The changes in the modularity Q of the collaboration networks of the eight
journals.
Journals
Change in modularity Q
Total (%)
By replication and
mutation (%)
By selection (%)
Journal of Communication .176 (100.0%) .075 (42.3%) .102 (57.6%)
Philosophy of Science .166 (100.0%) .074 (44.2%) .093 (55.8%)
Chemical Reviews .328 (100.0%) -.082 (-24.9%) .410 (124.9%)
Trends in Neurosciences .362 (100.0%) -.032 (-8.8%) .394 (108.8%)
Biometrika .315 (100.0%) .085 (26.8%) .231 (73.2%)
Journal of Theoretical Biology .456 (100.0%) .012 (2.7%) .444 (97.3%)
American Journal of Sociology .198 (100.0%) .066 (33.2%) .132 (66.8%)
Science .298 (100.0%) .000 (0.0%) .298 (100.0%)
5.4. The Estimation of Selection Bias
In the previous section, the influence of the selection mechanism on the change in
network connectedness was measured as the difference between the observed and
expected connectedness (i.e., 𝑧 𝑡 − 𝑧 ̂ 𝑡 ). From the viewpoint of conventional statistics, it is
merely the amount of the observed change unexplained by the replication and mutation
processes, that is, the residuals of the current growth model. However, the residuals of
the current model are expected to be consistently biased toward particular directions
because of the selection mechanism. To confirm this expectation, this section examines
the distribution of the percentile ranks of the observed connectedness using the
conventional methods of residual analysis.
153
The procedure involved two simple statistical tests (Figure 5.31). First, the
normality of the distribution of percentile ranks was tested using the Kolmogorov-
Smirnov test. The test statistics D quantifies the distance between the distribution of
empirical observations (i.e., the distribution of percentile ranks) and a reference
distribution (i.e., a normal distribution in this case). The statistical significance of this
statistics is calculated under the null hypothesis that the observations are drawn from the
reference distribution (i.e., 𝐻 𝑜 :𝐷 = 0). The Kolmogorov-Smirnov test allows us to judge
whether the residuals of the current model is normally distributed. Second, the average of
the percentile ranks of the observed connectedness, denoted by 𝑧 ∗
, was tested under the
null hypothesis that the average is equal to .50 (i.e., the median) using the one-sample t-
test (i.e., 𝐻 𝑜 :𝑧 ∗
= .50). If there is a substantial bias inherent in the selection of papers
and authoring teams, 𝑧 ∗
is significantly different from .50. Otherwise, 𝑧 ∗
is equal to .50.
Figure 5.31. The statistical tests for the estimation of selection bias.
The test results for the largest eigenvalues 𝜆 1
are summarized in Table 5.5. The
results of the Kolmogorov-Smirnov tests suggested that none of the percentile rank
154
distributions were significantly different from normal distributions (D = .056 to .178; p
= .068 to .384). On the other hand, the results of the one-sample t-tests suggested that the
average percentile ranks z* were significantly different from .50. This implies that
substantial pressures were imposed on the selection of papers and authoring teams and
such a tendency was consistent over years in all the eight journals. The average percentile
rank was particularly high in Trends in Neurosciences (z* = .772, p < .000). In the
journal, the observed networks were better connected than about 77% of the possible
networks on average. On the other hand, the average percentile rank was as low as .600
in the American Journal of Sociology (z* = .600, p < .000), meaning that the observed
networks were better connected than about 60% of the possible networks on average.
Therefore, the selection pressure was stronger in Trends in Neurosciences than in the
American Journal of Sociology.
Table 5.5. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests for the
percentile ranks of the observed largest eigenvalues λ1.
Journal
KS test One-sample t-test
D p z* 95% CI t df p
Journal of Communication .178 .078 .624 [.593 .655] 8.04 58 .000
Philosophy of Science .105 .369 .603 [.571 .636] 6.37 75 .000
Chemical Reviews .063 .270 .718 [.692 .743] 17.02 85 .000
Trends in Neurosciences .141 .555 .722 [.675 .769] 9.55 31 .000
Biometrika .057 .384 .670 [.642 .699] 12.03 108 .000
Journal of Theoretical Biology .057 .384 .712 [.681 .744] 13.58 48 .000
American Journal of Sociology .126 .152 .600 [.575 .626] 7.76 114 .000
Science .151 .068 .640 [.613 .666] 10.36 128 .000
155
The test results for spectral gap Δ𝜆 are summarized in Table 5.6. The results of
the Kolmogorov-Smirnov tests suggested that none of the percentile rank distributions
were significantly different from normal distributions (D = .047 to .172; p = .072
to .627). The results of the one-sample t-tests found the tendency that networks with low
values of Δ𝜆 were more likely to be observed than expected. However, the tendency was
statistically insignificant in the Journal of Communication (z* = .480, p = .300) and in
Philosophy of Science (z* = .475, p = .056). In contrast, the selection bias toward the low
levels of the spectral gap was the strongest in Trends in Neurosciences (z* = .262, p
< .000), suggesting that the observed networks had lower spectral gap than about 73% (=
1 – .262) of the alternative networks on average.
Table 5.6. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests for the
percentile ranks of the observed spectral gap Δλ.
Journal
KS test One-sample t-test
D P z* 95% CI t df p
Journal of Communication .057 .384 .480 [.441 .519] -1.05 58 .300
Philosophy of Science .047 .627 .475 [.450 .501] -1.94 75 .056
Chemical Reviews .157 .089 .318 [.298 .338] -17.90 85 .000
Trends in Neurosciences .172 .303 .262 [.228 .296] -14.10 31 .000
Biometrika .133 .072 .359 [.335 .382] -11.94 108 .000
Journal of Theoretical Biology .122 .456 .272 [.237 .307] -13.17 48 .000
American Journal of Sociology .117 .084 .383 [.358 .408] -9.20 114 .000
Science .125 .137 .402 [.377 .428] -7.48 127 .000
156
Finally, the test results for modularity Q are summarized in Table 5.7. The results
of the Kolmogorov-Smirnov tests suggested that none of the percentile rank distributions
were significantly different from normal distributions (D = .092 to .153; p = .057
to .806). The results of the one-sample t-tests suggested that networks with high
modularity were more likely to be observed than expected over time across the eight
journals. The selection bias toward high modularity was the strongest in Trends in
Neuroscience (z* = .646, p < .000). The observed networks had higher modularity than
about 65% of the possible networks on average in that journal.
Table 5.7. The results of the Kolmogorov-Smirnov tests and the one-sample t-tests for the
percentile ranks of the observed modularity Q.
Journal
KS test One-sample t-test
D p z* 95% CI t df p
Journal of Communication .153 .128 .527 [.510 .544] 3.21 58 .002
Philosophy of Science .145 .083 .523 [.506 .541] 2.61 75 .011
Chemical Reviews .116 .195 .594 [.566 .622] 6.64 85 .000
Trends in Neurosciences .125 .704 .646 [.603 .689] 6.91 31 .000
Biometrika .106 .173 .534 [.509 .559] 2.73 107 .007
Journal of Theoretical Biology .092 .806 .635 [.597 .673] 7.09 48 .000
American Journal of Sociology .127 .057 .518 [.505 .531] 2.66 113 .009
Science .116 .061 .522 [.510 .535] 3.45 128 .001
To summarize, a series of statistical tests found that the residuals of the current
model were normally distributed but their expected values were not equal to .50. This
implies that there were common and consistent biases inherent in all the observed
157
changes. All the results of the one-sample t-tests supported that the direction of the biases
was consistent with the expectation that collaboration networks with high values of the
largest eigenvalue and modularity and low values of spectral gap would be more likely to
be observed than others. In most cases the biases in the residuals were statistically
significant at a level of α = .05. But the biases were not significant in the Journal of
Communication and Philosophy of Science regarding spectral gap. Based on the test
results it can be drawn that the eight empirical collaboration networks had grown under
strong selection.
One interesting finding worth noting is that the biases toward three different
connectedness measures were highly correlated with each other (Figure 5.33). Despites
the small sample size (n = 8), all the pairwise correlation coefficients were statistically
significant. For example, the strongest selection pressures were found in Trends in
Neurosciences regarding all the three measures (the darkest circle on the bottom-right in
Figure 5.33), whereas the selection pressures were relatively weak in Philosophy of
Sciences and the Journal of Communication (on the up-left in Figure 5.33). The high
correlation among the estimated selection pressures for the three measures implies the
internal consistency among them.
158
Figure 5.32. The correlation among the selection biases toward the three network
connectedness measures. Note: the size of circles indicates the size of collaboration
networks in 2010, and the darkness of circles represents the strength of selection
pressures regarding modularity Q.
5.5. The Prediction of the Structural Change of Collaboration Networks
This section addresses the question about whether the current model is able to
reproduce the observed change in network connectedness of the empirical collaboration
networks. More specifically, the following questions were asked: (1) How well the model
fits the observed changes in the three connectedness measures? (2) To what degree the
replication and mutation processes and the selection mechanism explain the observed
change? To answer these questions, a regression analysis was conducted for each
measure and for each journal. The dependent variable was the observed change (∆𝑧 =
𝑧 𝑡 − 𝑧 𝑡 −1
), and the two predictor variables were (1) the expected change by the
replication and mutation processes (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
) and (2) the change due to the selection
159
mechanism (𝑧 𝑡 ∗
− 𝑧 ̂ 𝑡 ). The second predictor variable was obtained in three steps. First, for
example, the average percentile rank of the spectral gap of the observed networks of
Trends in Neurosciences was obtained as .262 between 1979 and 2010 (Table 5.6). Next,
the network with the 262nd smallest spectral gap in every year was sampled among the
1,000 alternative networks with the assumption that the selection pressure was constant
over time. Finally, the change due to the selection mechanism was measured by the
difference in spectral gap between the 262nd networks and that the expected changes.
The results of the regression analyses are summarized in Tables 5.8 to 5.10. In
each regression analysis, two regression models were tested. One model included the
change due to the replication and mutation processes only (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
). The other model
included both the change due to the replication and mutation processes (𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
) and
the change due to the selection mechanism (𝑧 𝑡 ∗
− 𝑧 ̂ 𝑡 ). By doing so, it was possible to
determine the degree to which the overall model was improved by adding the selection
mechanism (i.e., ΔR
2
).
First, the replication and mutation processes alone explained only a small
proportion of the variance in the change in the largest eigenvalue (R
2
= .056 to .163).
Nevertheless, all the models were found to be statistically significant at a significance
level of α = .05. The replication and mutation processes were positively associated with
the change in the largest eigenvalue (β = .236 to .357), meaning that the largest
eigenvalues of the collaboration networks increased even in the absence of selection
pressures. However, when the selection mechanism was included in the model, the
effects of the replication and mutation processes diminished. In the Journal of
Communication, for example, the regression coefficient of replication/mutation was
160
positive and significant (β = .236, t = 2.45, p < .000). But it became insignificant (β
= .119, t = 1.05, p = .300) when the selection mechanism was included in the model.
Also, the inclusion of the selection mechanism improved the model drastically (ΔR
2
= .270, ΔF = 23.81, p < .000).
The second set of regression analyses were conducted for the changes in spectral
gap. As presented in Table 5.9, the replication and mutation processes alone hardly
explained the observed change in spectral gap (R
2
= .007 to .056). In all the cases, the
regression coefficients of replication and mutation were statistically insignificant.
Instead, the observed change was mostly explained by the selection mechanism (R
2
= .085 to .592). The fact that all the regression coefficients of selection were negative
confirms the suppressing effects of the selection mechanism.
Finally, the effects of the replication and mutation processes and those of the
selection mechanism on the observed change in modularity were examined (Table 5.10).
As discussed in Section 5.3.4, the effects of the selection mechanism were positive and
statistically significant across all the journals, explaining a considerable amount of the
variance in the change in modularity (R
2
= .302 to .597). However, some mixed patterns
were found in the effects of the replication and mutation processes on modularity.
In the three journals—the Journal of Communication, Philosophy of Science, and
the American Journal of Sociology, the replication and mutation processes had positive
effects on the increase in modularity (β = .275, t = 2.16, p < .000; β = .262, t = 2.33, p
< .000; and β = .194, t = 2.10, p < .000, respectively). Those effects were still positive
and significant even after the selection process was included in the models (β = .256, t =
2.32, p < .000; β = .307, t = 3.12, p < .000; and β = .163, t = 2.08, p < .000, respectively).
161
In those journals, all the three evolutionary mechanisms contributed to the increase in
modularity. On the other hands, the other five journals—Chemical Reviews, Trends in
Neurosciences, Biometrika, the Journal of Theoretical Biology, and Science, the
replication and mutation processes had negative effects on the increase in modularity,
although the effects were not statistically significant (β = -.202, t = -1.89, p = .063; β =
-.244, t = -1.38, p = .177; β = -.124, t = -1.29, p = .052; β = -.085, t = -0.59, p < .559, and
β = -.101, t = -1.15, p = .254 respectively).
Overall, the regression analyses found that the observed structural change of the
collaboration networks was well explained by the three evolutionary mechanisms—
replication, mutation, and selection. In particular, the results showed that the selection
process was an important predictor of the evolutionary change of collaboration networks.
In most cases, the replication and mutation processes alone did not sufficiently account
for the variance in observed change. Even when they produced statistically significant
results, their explanatory power was unsatisfactory. However, when the effects of the
selection mechanism were taken into account, the models were significantly improved.
162
Table 5.8. Regression analyses for the observed change in the largest eigenvalue as the
dependent variable.
Journal
Replication/
Mutation
Selection R
2
F ΔR
2
ΔF
Journal of
Communication (N = 59)
.309 (2.45)
***
.095 6.00
.119 (1.05)
.553 (4.88)
***
.365 16.10
***
.270 23.81
***
Philosophy of Science
(N = 76)
.254 (2.26)
***
.065 5.11
***
.226 (2.34)
***
.507 (5.25)
***
.321 17.26
***
.256 27.66
***
Chemical Reviews
(N = 86)
.404 (4.04)
***
.163 16.40
***
.300 (4.41)
***
.688 (10.10)
***
.625 17.26
***
.462 102.31
***
Trends in Neurosciences
(N = 32)
.357 (2.10)
***
.127 4.37
***
.046 (0.36)
.762 (6.02)
***
.612 22.85
***
.485 36.27
***
Biometrika
(N = 109)
.236 (2.51)
***
.056 6.32
***
.117 (1.60)
.638 (8.70)
***
.450 43.20
***
.393 75.71
***
Journal of Theoretical
Biology (N = 49)
.324 (2.36)
***
.105 5.51
***
.021 (0.21)
.778 (7.88)
***
.619 37.28
***
.514 62.11
***
American Journal of
Sociology (N = 115)
.244 (2.68)
***
.060 7.18
***
.137 (1.82)
.582 (7.73)
***
.389 35.32
***
.327 59.70
***
Science
(N = 129)
.277 (3.25)
***
.077 10.68
***
.216 (3.29)
***
.622 (9.46)
***
.460 53.71
***
.383 89.52
***
* p < .05, ** p < .01, *** p < .001
t-statistics in parentheses
N is the number of years of observation
163
Table 5.9. Regression analyses for the observed change in spectral gap as the dependent
variable.
Journal
Replication/
Mutation
Selection R
2
F ΔR
2
ΔF
Journal of
Communication (N = 59)
.153 (1.17)
.024 1.37
.113 (0.87)
-.252 (-1.97)
*
.086 2.61
.062 3.89
*
Philosophy of Science
(N = 76)
.087 (0.75)
.007 0.57
.105 (0.96)
-.343 (-3.13)
***
.125 5.23
***
.118 9.81
***
Chemical Reviews
(N = 86)
.074 (0.68)
.005 0.46
.054 (0.76)
-.766 (-10.90)
***
.592 60.33
***
.587 120.03
***
Trends in Neurosciences
(N = 32)
.084 (0.46)
.127 0.21
.018 (0.14)
-.739 (-5.90)
***
.549 17.63
***
.542 34.81
***
Biometrika
(N = 109)
.120 (1.25)
.056 1.55
.081 (1.08)
-.623 (-8.26)
***
.401 35.41
***
.386 68.30
***
Journal of Theoretical
Biology (N = 49)
.216 (1.51)
.047 2.29
.126 (1.20)
-.681 (-6.49)
***
.503 23.23
***
.456 42.19
***
American Journal of
Sociology (N = 115)
.182 (1.97)
.033 3.86
.101 (1.25)
-.519 (-6.46)
***
.295 23.48
***
.262 41.69
***
Science
(N = 129)
.136 (1.54)
.018 2.38
***
.093 (1.32)
*
-.600 (-8.50)
***
.376 37.96
***
.358 72.23
***
* p < .05, ** p < .01, *** p < .001
t-statistics in parentheses
N is the number of years of observation
164
Table 5.10. Regression analyses for the observed change in modularity as the dependent
variable.
Journal
Replication/
Mutation
Selection R
2
F ΔR
2
ΔF
Journal of
Communication (N = 59)
.275 (2.16)
***
.076 4.65
***
.256 (2.32)
***
.496 (4.51)
***
.321 13.26
***
.246 20.31
***
Philosophy of Science
(N = 76)
.262 (2.33)
***
.068 5.43
***
.307 (3.12)
***
.485 (4.94)
***
.302 15.75
***
.233 24.40
***
Chemical Reviews
(N = 86)
.-.202 (-1.89)
.041 3.56
-.130 (-1.70)
.692 (8.99)
***
.514 43.86
***
.473 80.76
***
Trends in Neurosciences
(N = 32)
-.244 (-1.38)
.060 1.90
-.230 (-1.94)
.733 (6.22)
***
.597 21.52
***
.538 38.66
***
Biometrika
(N = 109)
-.124 (-1.29)
.015 1.67
-.149 (-1.95)
.616 (8.14)
***
.394 34.51
***
.379 66.25
***
Journal of Theoretical
Biology (N = 49)
-.085 (-0.59)
.007 0.35
-.146 (-1.45)
.787 (7.81)
***
.573 30.88
***
.566 61.00
***
American Journal of
Sociology (N = 115)
.194 (2.10)
***
.038 4.43
***
.163 (2.08)
***
.529 (6.77)
***
.317 26.00
***
.279 45.81
***
Science
(N = 129)
-.101 (-1.15)
.010 1.31
***
-.067 (-0.96)
.610 (8.70)
***
.382 38.87
***
.371 75.67
***
* p < .05, ** p < .01, *** p < .001
t-statistics in parentheses
N is the number of years of observation
165
In the above regression analyses the three connectedness measures—the largest
eigenvalue, spectral gap, and modularity—have been examined separately. However, it is
worth noting that they measure different aspects of the structure of a network and are not
completely independent of each other. Therefore, the question to be address would
become whether or not the network sampled from all the possible networks to predict the
value of one measure of the observed network would also predict the value of another
measure fairly well. For example, the network with the 262nd smallest spectral gap
among the 1000 possible networks well predicted the spectral gap of the observed
collaboration network of Trends in Neuroscience. Then, the largest eigenvalue and
modularity of the sampled network should also predict those of the observed network as
well, if the selection mechanism indeed operates.
To address this question, all the possible networks generated for the collaboration
network of Trends in Neuroscience were reexamined. First, the differences in spectral
gap between all the possible networks and the observed network were computed,
standardized by dividing by the standard deviation of the differences, and denoted by
𝑑 Δ𝜆 . In the same manner, the standardized difference in modularity between all the
possible networks and the observed network were obtained and denoted by 𝑑 𝑄 . Figure
5.33 presents the scatter plot of 𝑑 Δ𝜆 and 𝑑 𝑄 with the marginal frequency distributions
(𝑁 = 32,000 = 1,000× 32 years). The average of 𝑑 Δ𝜆 was .692 (the blue vertical line),
whereas that of 𝑑 𝑄 was -.511 (the blue horizontal line), implying that the possible
networks as a whole poorly predicted the spectral gap and modularity of the observed
network (the black vertical and horizontal lines).
166
According to the results in Table 5.5, the estimated average percentile rank for the
largest eigenvalue was .722 with the 95% confidence interval of [.675, .769]. For every
year, the 95 (= 769 – 675 + 1) possible networks whose percentile ranks for the largest
eigenvalue fell into the interval were sampled and are presented as red dots on the scatter
plot (𝑛 = 3,040 = 95 × 32 years). The average of 𝑑 Δ𝜆 of the sampled networks
was .014 (the red vertical line), whereas the average of 𝑑 𝑄 of the sampled networks
was .111 (the red horizontal line), implying that the sampled networks were much better
at predicting both the spectral gap and modularity of the observed network. Welch’s two-
sample t-tests
13
confirmed that the sampled networks were significantly more similar to
the observed networks in terms of spectral gap and modularity than all the possible
networks were (t = -63.23, df = 6806.75, p < .000; t = 67.74, df = 6493.00, p < .000). This
suggests that the networks sampled to predict the largest eigenvalue of the observed
network fairly well predicted the spectral gap and modularity of the observed network as
well.
This procedure was repeated to answer the questions about (1) how well the
networks sampled to predict the spectral gap of the observed network predicted the
modularity and largest eigenvalue of the observed network (Figure 5.34) and (2) how
well the networks sampled to predict the modularity of the observed network predicted
the largest eigenvalue and spectral gap of the observed network (Figure 5.35). In both
cases, the sampled networks were significantly more similar to the observed networks
than all the possible networks were.
13
The null hypothesis was that there was no differences in standardized difference between the sampled
networks and all the possible networks.
167
Figure 5.33. The scatter plot of the standardized differences in spectral gap and
modularity between the possible and observed networks with marginal frequency
distributions.
168
Figure 5.34. The scatter plot of the standardized differences in modularity and the largest
eigenvalue between the possible and observed networks with marginal frequency
distributions.
169
Figure 5.35. The scatter plot of the standardized differences in the largest eigenvalue and
spectral gap between the possible and observed networks with marginal frequency
distributions.
170
5.6. Further Analysis with Modified Team Formation Mechanisms
The results presented in the previous sections suggest that the growth mechanism
by the replication and mutation processes alone did not fully explain the structural
changes of the empirical collaboration networks. It was found that the structure of the
observed networks consistently deviated from the expected structure, which was
interpreted as the effects of the selection mechanism operated on papers and authoring
teams. However, the consistent discrepancies between the expected and observed
structure could be attributed to other sources than the selection mechanism. One possible
source would be systematic biases embedded in the team formation mechanism. The
current team formation mechanism assumes that every existing authoring team is equally
likely to be sampled for replication (i.e., uniform team sampling). In reality, however,
more recent collaborations would be more likely to be repeated than old ones. If that is
case, the uniform team sampling for replication could introduce a systematic bias, in
particular, if team sizes change over the years, resulting in consistent biases in estimating
the structural change of the empirical collaboration networks.
To address this problem, the team formation mechanism was modified such that
recent teams are more likely than old teams to be sampled for replication. In the modified
mechanism, the probability of an authoring team being sampled for replication was
defined as a function of its age [i.e., 𝑓 (𝑦 ), where y is the age of team]. More specifically,
an exponential probability function 𝑓 (𝑦 ) = 𝑐 ∙ 𝑒 −𝑦 and a polynomial function 𝑓 (𝑦 ) =
𝑐 (1+𝑦 )
2
were used to describe the sampling bias toward recent teams, where c is a
normalizing constant in each function. The modified team formation mechanisms were
applied to the collaboration network of Trends in Neurosciences, and the structural
171
change of the network was reanalyzed. The structural properties of the networks were
measured by the largest eigenvalue, spectral gap, and modularity in the same way as
before in order to compare the new results with the previous results.
As summarized from Figures 5.36 to 5.41, the new results were almost identical
to the previous results in Section 5.3. The observed largest eigenvalues were almost
always greater than the expected largest eigenvalues (Figures 5.36 and 5.37), the
observed spectral gaps were almost always less than the expected spectral gaps (Figures
5.38 and 5.39), and the values of observed modularity were almost always greater than
the expected values (Figures 5.40 and 5.41). These results together imply that the uniform
team formation model was as good (or bad) at predicting the observed structural change
as the modified models in which recent teams are more likely to be sampled. Further, it is
less likely that the consistent discrepancies between the expected and observed structure
of collaboration networks were due to the systematic biases embedded in the team
formation mechanism.
172
(a)
(b)
Total change:
991.91 (100.0%)
By
replication/mutation:
602.71 (60.8%)
By selection:
308.20 (39.2%)
Figure 5.36. The observed and expected changes in the largest eigenvalue λ1 of the
collaboration network of Trends in Neurosciences with the exponential team sampling.
(a)
(b)
Total change:
991.91 (100.0%)
By
replication/mutation:
601.38 (60.6%)
By selection:
509.53 (39.4%)
Figure 5.37. The observed and expected changes in the largest eigenvalue λ1 of the
collaboration network of Trends in Neurosciences with the polynomial team sampling.
173
(a)
(b)
Total change:
200.83 (61.1%)
By
replication/mutation:
328.73 (100.0%)
By selection:
-127.90 (-38.9%)
Figure 5.38. The observed and expected changes in the spectral gap Δ𝜆 of the
collaboration network of Trends in Neurosciences with the exponential team sampling.
(a)
(b)
Total change:
200.83 (60.3%)
By
replication/mutation:
333.12 (100.0%)
By selection:
-132.29 (-39.7%)
Figure 5.39. The observed and expected changes in the spectral gap Δ𝜆 of the
collaboration network of Trends in Neurosciences with the polynomial team sampling.
174
(a)
(b)
Total change:
.362 (100.0%)
By
replication/mutation:
-.030 (-8.3%)
By selection:
.392 (108.3%)
Figure 5.40. The observed and expected changes in the modularity Q of the collaboration
network of Trends in Neurosciences with the exponential team sampling.
(a)
(b)
Total change:
.362 (100.0%)
By
replication/mutation:
-.033 (-9.1%)
By selection:
.395 (109.1%)
Figure 5.41. The observed and expected changes in the modularity Q of the collaboration
network of Trends in Neurosciences with the polynomial team sampling.
175
In the same way as before a series of regression analyses was performed using the
connectedness measures of the newly generated networks by the modified team
formation mechanisms. As presented in Table 5.11, the results were almost identical to
the previous results in Section 5.5. In all the cases, the observed changes in the
connectedness measures were better explained when both the effects of the replication
and mutation processes and those of the selection mechanism were included in the model.
In addition, the spectral gap of the collaboration networks was re-compute using
the random walk matrices of the networks instead of the adjacency matrices, which is
denoted by ∆𝜆 𝐏 . The random walk matrix of a network P is the matrix with elements 𝑝 𝑖𝑗
such that
𝑝 𝑖𝑗
= {
1
deg
𝑜𝑢𝑡 (𝑖 )
if nodes 𝑖 and 𝑗 are connected
0 otherwise,
where deg
𝑜𝑢𝑡 (𝑖 ) is the number of the outgoing edges from node i. The advantage of
using random walk matrices in computing the spectra of networks is that the eigenvalues
are between -1 and 1, and the largest eigenvalue 𝜆 1
is always equal to 1, so they
normalize out dependencies on node degrees.
The changes in the normalized spectral gap ∆𝜆 𝐏 of the collaboration network of
Trends in Neurosciences were examined. In that case, the expected spectral gaps were
obtained from the possible networks generated by the team formation mechanisms with
(1) uniform team sampling, (2) exponential team sampling, and (3) polynomial team
sampling.
176
Table 5.11. Regression analyses for the observed change in the three connectedness
measures of the collaboration network of the Trends in Neurosciences as the dependent
variables (N = 32).
(a) Team formation mechanism with exponential team sampling
Dependent variable
Replication
/Mutation
Selection R
2
F ΔR
2
ΔF
The largest eigenvalue
.472 (2.93)
***
.223 8.60
***
.232 (1.91)
.679 (5.60)
***
.627 24.33
***
.404 31.36
***
Spectral gap
.098 (0.54)
.010 0.29
.049 (0.38)
-.704 (-5.37)
***
.503 14.69
***
.493 28.82
***
Modularity
-.302 (-1.74)
.091 3.02
-.224 (-1.74)
.663 (5.14)
***
.516 15.99
***
.433 26.41
***
(b) Team formation mechanism with polynomial team sampling
Dependent variable
Replication
/Mutation
Selection R
2
F ΔR
2
ΔF
The largest eigenvalue
.446 (2.72)
***
.198 7.43
***
.236 (1.97)
.690 (5.82)
***
.630 24.76
***
.432 33.93
***
Spectral gap
.105 (0.58)
.011 0.33-
.095 (0.70)
-.732 (-5.40)
***
.507 14.09
***
.496 29.16
***
Modularity
-.293 (-1.68)
.086 2.82
-.150 (-1.16)
.686 (5.30)
***
.536 16.72
***
.449 28.06
***
* p < .05, ** p < .01, *** p < .001;
t-statistics in parentheses
177
The results suggest that the normalized spectral gap of the collaboration network
had substantially increased over time (Figures 5.42 to 5.44), although the increasing
patterns were different from those of the increase in the non-normalized spectral gap.
Also, the suppressing effects of the selection mechanism were observed, suggesting that
the selection mechanism had operated in the opposite direction of the change due to the
replication and mutation processes. More specifically, about 25% of the increased
normalized spectral gap by the replication and mutation processes was reduced by the
selection mechanism.
(a)
(b)
Total change:
.0136 (74.3%)
By
replication/mutation:
.0183 (100.0%)
By selection:
-.0047 (-25.7%)
Figure 5.42. The observed and expected changes in the normalized spectral gap ΔλP of
the collaboration network of Trends in Neurosciences with the uniform team sampling.
178
(a)
(b)
Total change:
.0136 (76.0%)
By
replication/mutation:
.0179 (100.0%)
By selection:
-.0043 (-24.0%)
Figure 5.43. The observed and expected changes in the normalized spectral gap ΔλP of
the collaboration network of Trends in Neurosciences with the exponential team
sampling.
(a)
(b)
Total change:
.0136 (75.6%)
By
replication/mutation:
.0180 (100.0%)
By selection:
-.0044 (-24.4%)
Figure 5.44. The observed and expected changes in the normalized spectral gap ΔλP of
the collaboration network of Trends in Neurosciences with the polynomial team
sampling.
179
The relative importance of the replication and mutation processes and the
selection mechanism in explaining the observed change in normalized spectral gap was
examined by performing a series of regression analyses. As presented in Table 5.12, the
replication and mutation processes alone hardly explained the observed change in
normalized spectral gap (R
2
= .007 to .056). In all the cases, the regression coefficients of
replication and mutation were statistically insignificant. However, when the selection
mechanism was included, the regression models were much improved (R
2
= .486 to .518).
The results of the regression analyses were comparable to the previous results in Section
5.5.
Table 5.12. Regression analyses for the observed change in the normalized spectral gap
of the collaboration network of Trends in Neurosciences as the dependent variable (N =
32).
Team sampling method
Replication
/Mutation
Selection R
2
F ΔR
2
ΔF
Uniform sampling
.080 (0.44)
.006 0.19
.182 (1.41)
-.727 (-5.63)
***
.525 16.02
***
.518 31.64
***
Exponential sampling
.087 (0.48)
.008 0.23
.053 (0.40)
-.692 (-5.19)
***
.485 13.67
***
.478 26.90
***
Polynomial sampling
.107 (0.59)
.012 0.35
.103 (0.79)
-.697 (-5.29)
***
.497 14.33
***
.486 28.00
***
* p < .05, ** p < .01, *** p < .001
t-statistics in parentheses.
180
CHAPTER 6: DISCUSSION AND CONCLUSIONS
Scientific collaboration networks grow and change as new nodes and edges are
added. However, the addition of nodes and edges is highly selective in collaboration
networks unlike other networks. Not all nodes can join the networks or to freely form
edges to other nodes in the networks. Only authors whose papers are selected for
publication can join the network and be connected only to their coauthors. The selection
of papers for publication determines the set of nodes and edges to be added and
consequently the direction of the structural change of collaboration networks. This unique
characteristic of collaboration networks requires us to develop a new model of network
growth that can account for the selection mechanism. From this perspective, the
dissertation proposed an evolutionary model of collaboration networks based on the three
fundamental mechanisms—replication, mutation, and selection—and showed that the
model well predicted the direction of the structural change real-world collaboration
networks.
6.1. Summary of Findings
6.1.1. The Proposed Model
The model proposed in the dissertation incorporates the three fundamental
mechanisms of evolutionary dynamics—replication, mutation, and selection. The
replication and mutation processes were adopted to formulate the growth mechanism of
collaboration networks. The growth mechanism describes the process in which new
authoring teams are formed by replicating existing teams with mutation and generates a
whole ensemble of possible networks by adding the new authoring teams to an ancestral
network. Selection occurs when papers with particular knowledge elements are more
181
likely than others to be selected for publication, and thus, when authors who are capable
to produce acceptable papers either by themselves or through collaboration are more
likely to be selected to be added to existing networks. The selection mechanism has
significant impacts on the structural properties of collaboration networks. More
specifically, when the range of acceptable papers is narrow (i.e., under strong selection),
collaboration networks tend to show high values of the largest eigenvalue 𝜆 1
and
modularity Q and low values of spectral gap Δ𝜆 .
Under the framework of the current model, a collaboration network observed at a
given point in time t is one of many possible networks that stem from its ancestral
network observed at the previous point in time t – 1. If collaboration networks grow by
the replication and mutation processes in the absence of selection pressure, a structural
property of the observed network at t is expected to be equal to that of the ensemble of
the possible networks (i.e., 𝑧 𝑡 = 𝑧 ̂ 𝑡 ). Therefore, the difference in the structural property
between the ancestral network and the ensemble indicates the amount of the change due
to the replication and mutation processes (i.e., 𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
). When the selection mechanism
operates on papers and authoring teams, the observed structural property at t is different
from the expected property, in particular, to the extent to which the selection is strong.
Therefore, the difference between the observed and expected structural properties
indicates the direction and degree of the selection pressure (i.e., 𝑧 𝑡 − 𝑧 ̂ 𝑡 ).
This logic allows us to analyze the structural change of collaboration networks
between different points in time (i.e., ∆𝑧 = 𝑧 𝑡 − 𝑧 𝑡 −1
) by decomposing it into two parts
according to the sources of the change. First, collaboration networks grows as new
authoring teams are added, which possibly (but not necessarily) results in the structural
182
change of the networks. This change can be explained by the replication and mutation
processes in the team formation mechanism and measured by the difference in structural
properties between the expected network at one point in time and the ancestral network at
the previous point in time (i.e., 𝑧 ̂ 𝑡 − 𝑧 𝑡 −1
). Second, because not all authoring teams are
equally capable to produce acceptable papers, or because not all papers are accepted for
publication, some teams are more likely than others to be selected to be added. The
selection of authoring teams (or their papers) therefore leads to further structural change
than expected by the pure growth mechanism. This additional change can be explained by
the selection mechanism and measured by the difference in structural properties between
the observed and expected networks at time t (i.e., 𝑧 𝑡 − 𝑧 ̂ 𝑡 ).
In a sense, the effects of the selection mechanism on the structural change of
collaboration networks can be simply treated as the amount of the change unexplained by
the growth mechanism, namely, the residuals of the growth model. However, the
residuals would not be evenly distributed around zero, as long as a selection mechanism
operates consistently during the evolution of collaboration networks. Especially when the
structural property of interest is network connectedness, the residuals should be biased
toward high levels of the largest eigenvalue 𝜆 1
and modularity Q but low levels of
spectral gap Δ𝜆 .
6.1.2. The Evolutionary Change of Empirical Collaboration Networks
The proposed model was tested against empirical collaboration networks. The
empirical networks were constructed based on the bibliographic information of the papers
published in the eight leading journals in their fields across disciplines: (1) the Journal of
Communication, (2) Philosophy of Science, (3) Chemical Reviews, (4) Trends in
183
Neurosciences, (5) Biometrika, (6) the Journal of Theoretical Biology, (7) the American
Journal of Sociology, and (8) Science.
For each journal, the changes in three network connectedness measures were
examined. All the three measures increased over time across all the networks. The total
amount of the observed changes in network connectedness was decomposed into (1) the
amount of the change due to the replication and mutation processes and (2) the amount of
the change due to the selection mechanism. The decomposition of the observed changes
revealed interesting patterns in the relative contributions of the replication and mutation
processes and the selection mechanism to explaining the structural changes of the
collaboration networks.
The replication and mutation processes increased the largest eigenvalues of the
collaboration networks. This means that the ability of collaboration networks to remain
connected under random deletion of nodes or edges could be improved simply by the
growth mechanism. This finding is consistent with previous studies on network
robustness (Albert et al., 2000). Albert at al. (2000) found that simple preferential
attachment processes can sufficiently improve the tolerance of networks against random
errors. Recall that the preferential attachment process is the simplest form of edge
replication (Section 2.4.1) and also embedded in the team formation mechanism of the
current model (Section 4.3). However, it should be noted that the selection mechanism
further increased the largest eigenvalue to a considerable degree, meaning that real-world
networks were much more robust than expected by simple growth mechanisms. The
regression analyses found that the effects of the replication and mutation processes on the
increase in the largest eigenvalues were less important than those of the selection
184
mechanism. Moreover, the variance in the observed changes was better explained when
the selection mechanism was included in the model (Table 5.8).
The replication and mutation processes increased the spectral gap of the
collaboration networks, meaning that new edges tended to be formed between clusters
rather than within clusters. To demonstrate, consider a network that is composed of
several cohesive clusters, which are loosely connected by a few bridges (e.g., Figure
3.13b). Suppose that a pair of authors collaborate to publish a paper. If they are from the
same cluster, it is likely that they collaborated before so that their current collaboration is
repeated one that does not create a new edge or alter the network structure. On the other
hand, if they are from different clusters, it is unlikely that they collaborated before so that
their collaboration is likely to form a new edge, in particular, a bridge connecting the
clusters to which the authors belong.
However, the selection mechanism operated in the opposite direction, lowering
the increased spectral gap due to the replication and mutation processes. Put differently,
less inefficient networks were more likely than more inefficient ones to be realized, as the
results of the selection mechanism. Although the observed spectral gap increased over
time because the suppressing effects of the selection mechanism were not as large as the
effects of the replication and mutation processes, the selection mechanism surely
operated to lower the level of spectral gap. The regression analyses suggested that the
observed structural properties could be better explained when all the three evolutionary
mechanisms—replication, mutation, and selection—were included in the model (Table
5.9).
185
The effects of the replication and mutation processes on the change in network
modularity were inconsistent across the eight collaboration networks. The two processes
increased network modularity when the collaboration networks were sparse but decreased
network modularity when the collaboration networks were dense. This finding is in
agreement with previous studies that showed that random networks display intrinsic
modularity and may be partitioned yielding high values, especially when the networks are
sparse (Guimerà et al., 2004; Reichardt & Bornholdt, 2006). Nevertheless, the modularity
of the collaboration networks increased over time largely due to the selection mechanism,
which supports the expectation that collaboration networks grow modular over time
under strong selection.
A remarkably consistent pattern observed across all the eight collaboration
networks, which deserves attention, is that the selection mechanism was an important
factor for understanding the structural change of collaboration networks. The residual
analysis provided strong evidence showing that the effects of the selection mechanism on
the structural change of the collaboration networks were not merely the fitting errors of
the growth model. A common and consistent bias was observed in the residuals, and
more importantly, the bias was oriented toward the direction expected by the selection
model. In addition, a large proportion of the variance in the observed structural change
was explained when the effects of the selection mechanism were included in the model,
whereas the replication and mutation processes had minor or even insignificant effects on
the change. Overall, the results of the current analysis suggests that the structural change
of collaboration networks can be better explained by the evolutionary model that
186
incorporates all the three fundamental mechanisms—replication, mutation, and
selection—rather than simple growth mechanisms.
6.2. Limitations and Directions for Further Research
The evolutionary model of collaboration networks proposed in the dissertation
provides important insights into understanding of the growth dynamics of the empirical
collaboration networks. However, this study contains a number of limitations that suggest
meaningful directions for future research.
First, the dissertation conceptualized the strength of selection pressure on
collaboration networks as the range of acceptable papers by journals, namely, the
journals’ sweet spots, and established the relationship between selection pressure and
network connectedness. The paper production and acceptance process introduced in the
dissertation is based upon the well-developed model used in previous studies to explore
the emergence of scientific communities (Gilbert, 1997; Guimerà et al., 2005). In
addition, the simulation results were intuitive and compatible with the findings of
previous research (Clune et al., 2013; Demetrius & Manke, 2005; Estrada, 2006).
However, it should be noted that the relationship between selection pressure and network
connectedness was drawn from a rather simple model that may not fully reflect the reality
of the publication decision processes. For example, the current model assumes that every
author possesses the same number of knowledge elements and does not change their
knowledge elements. However, authors can learn new knowledge, in particular, from
their collaborators. Accordingly, their knowledge elements can change over time. For this
problem, further elaboration of the paper production and acceptance process is required,
187
and the influences on the relationship drawn from the current model needs to be
reexamined either numerically or analytically.
The second and related issue is concerned with the interpretation of the practical
meaning of selection pressure, in particular, regarding the quality of journals. The results
suggested that some journals had stronger selection pressures (e.g., Chemical Reviews,
Trends in Neurosciences, and the Journal of Theoretical Biology) than others (e.g., the
Journal of Communication, Philosophy of Science, and Science). One interesting and
practical question would be whether journals with strong selection pressures are “better”
than those with weak selection pressures. However, the current findings cannot directly
answer this question, because all the eight academic journals included in the analysis
were the leading journals in their fields. Each journal must be functional for their unique
goals and purposes. Note that the strength of selection pressure was conceptually
measured by the range of acceptable papers. Thus, journals with strong selection
pressures are simply those that are highly specialized in well-defined research areas,
whereas journals with weak selection pressures are those that cover general and broad
research topics.
Third, the empirical collaboration networks were constructed based on the
collaborative relations observed only in the sampled journals, which may conceal the true
relations among authors. For example, some authors who appear as to be disconnected in
the current networks might have well-established relations which were built through their
collaboration in other journals. If they collaborate in the sampled journals, their relations
will be treated as new collaboration links resulting from mutation, but the relations are
actually repeated one resulting from replication. In that case, the replication and mutation
188
processes may not fully capture the true team formation mechanism. More specifically,
the impacts of mutation processes could be overestimated. This problem is similar to that
of the boundary specification in traditional social network analysis but easier to solve.
The MAS database provides a set of APIs that make it possible to collect all the papers
that individual authors have written, which were used to measure authors’ professional
lifetimes in this study. Utilizing the bibliographic information of those papers, it is
possible to identify authors’ complete collaborative relations other than in the sampled
journals. By doing so, the team formation mechanism and its impacts on network
structure can be estimated more accurately.
Fourth, although the current model treats authoring teams as a basic unit of social
and communication relations among authors, it does not include any variables that
characterize the teams, which may be directly related to their ability to produce
acceptable papers. For example, team diversity would be an important predictor variable,
because it has been found as to be one of the most significant variables to explain team
performance in previous studies (Bercovitz & Feldman, 2011; Freeman & Huang, 2014;
Lungeanu & Contractor, 2015; van Knippenberg & Schippers, 2007). In the current
context, team diversity can be measured in three different ways: (1) diversity in
publication experience (i.e., 𝑞 𝑖 ), (2) diversity in collaboration experience (i.e., 𝑘 𝑖 ), and (3)
diversity in expertise domain. In particular, the third variable can be measured as the
number of different disciplines in which team members have published papers utilizing
the bibliographic information of all the papers they have written. By incorporating those
team-level characteristics, the selection mechanism can be more fully understood.
Further, the results from the model that includes those variables will provide more
189
practical implications for team performance, answering the question about whether and in
what way the diversity of teams is related to their success in publishing papers.
Final and most important, the current model and analysis assumed that the
selection mechanism is constant and consistent over time, excluding the possibility that
the selection mechanism itself could change. This assumption made it easy to detect and
estimate the effects of the selection mechanism. However, it is more reasonable to expect
that journals’ selection criteria change over time, especially when we consider the rapidly
changing academic environment. Before the 1990’s, for instance, communication via
computers or other digital media was not a preferred research topic for the Journal of
Communication. At that time, most published papers focused on the traditional face-to-
face communication or communication through mass media. However, the past decade
has witnessed the rapid growth of computer-mediated-communication research in the
journal, implying the change in the selection criteria of the journal, at least, regarding
preferred research topics. The change in journals’ selection criteria is expected to have
nontrivial impacts on the evolution of collaboration networks. Therefore, the possibility
of varying criteria should be taken into account to elaborate the current model. The
elaborated model would requires more sophisticated analytical methods, such as time-
series analysis.
6.3. Conclusions
A major contribution of the dissertation is that it provides a new perspective to the
study of social and communication networks by combining the two powerful theoretical
frameworks—network theory and evolutionary theory. Network theory helps to
understand the complexity inherent in the emerging patterns of collaborative relations
190
among researchers, while evolutionary theory provides important insights into explaining
the growth dynamics of the networks. To our knowledge, this is the very first study that
incorporates all the three evolutionary mechanisms—replication, mutation, and
selection—into the model of network growth and demonstrates the effects of selection
mechanism on the structural change of networks.
The current model was developed to capture the evolutionary dynamics of
scientific collaboration networks, focusing on their unique characteristic, namely, the
selective addition of nodes and edges. However, its application does not have to be
limited to scientific collaboration networks. Selection can take place in different forms in
different kinds of networks, and thus, can be formulated in different ways. Perhaps, one
of the most general forms of selection in network evolution would be the selective
deletion of nodes and edges. Network structure changes not just when new nodes and
edges are added but also when existing node and edges are deleted. Then, selection can
occurs when nodes or edges are deleted at different rates depending on their different
characteristics. This form of selection leads first to the decline and eventually the
dissolution of networks. Hence, the understanding of the selective deletion mechanism
will lead us to develop a theory that describes the whole life cycle of networks: the birth,
growth, decline, and death of networks. In the dissertation, the small first step was taken
toward the theory.
191
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APPENDICES
Appendix 1. The Derivation of the Price Equation
A1.1. Statement
Consider a population of n individuals, each of which has a heritable trait
described by a value of z. Let subscript i identify the group with trait 𝑧 𝑖 and let 𝑤 𝑖 be the
growth rate of that group. The Price equation states that the change in the average trait of
the population from one generation to the next generation, denoted by Δ𝑧 ̅ = 𝑧 ̅
′
− 𝑧 ̅ , is
decomposed into two parts:
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 Δ𝑧 𝑖 )
𝑤̅
,
(A.1)
where ∆𝑧 𝑖 is the change in character value for the children of group i and defined as 𝑧 𝑖 ′
−
𝑧 𝑖 . The first term, the covariance between growth rate and character value,
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )/𝑤̅, gives the amount of change due to selection. The second term,
𝐸 (𝑤 𝑖 Δ𝑧 𝑖 )/𝑤̅, is the amount of change due to the trait transmission.
A1.2. Mean and Covariance
To derive the Price equation the definitions of mean and covariance are
necessary. The mean of 𝑥 𝑖 that occurs 𝑛 𝑖 times in a set of n elements (𝑛 = ∑ 𝑛 𝑖 𝑖 ) is
𝑥 ̅ = E(𝑥 𝑖 ) ≝
1
∑𝑛 𝑖 𝑖 ∑ 𝑥 𝑖 𝑛 𝑖 𝑖 =
1
𝑛 ∑ 𝑥 𝑖 𝑛 𝑖 𝑖 .
The covariance between 𝑥 𝑖 and 𝑦 𝑖 both of which occur 𝑛 𝑖 times is
cov(𝑥 𝑖 ,𝑦 𝑖 ) ≝
1
∑𝑛 𝑖 𝑖 ∑ 𝑛 𝑖 [𝑥 𝑖 − E(𝑥 𝑖 )][𝑦 𝑖 − E(𝑦 𝑖 )]
𝑖 = E(𝑥 𝑖 𝑦 𝑖 )− E(𝑥 𝑖 )E(𝑦 𝑖 ).
214
A1.3. Derivation
Let 𝑛 𝑖 be the number of individuals with trait 𝑧 𝑖 in one generation (i.e., parents)
and 𝑛 𝑖 ′
be the number of their offspring in the next generation (i.e., children). The total
number of parents is 𝑛 = ∑ 𝑛 𝑖 𝑖 , and the number of children is 𝑛 ′
= ∑ 𝑛 𝑖 ′
𝑖 . The grow rate
of group i is defined as the ratio of children with trait 𝑧 𝑖 to their parents:
𝑤 𝑖 =
𝑛 𝑖 ′
𝑛 𝑖 . (A.2)
The average growth rate of the whole population is
𝑤̅ = E(𝑤 𝑖 ) ≝
1
∑𝑛 𝑖 𝑖 ∑ 𝑤 𝑖 𝑛 𝑖 𝑖 =
1
𝑛 ∑
𝑛 𝑖 ′
𝑛 𝑖 𝑛 𝑖 𝑖 =
1
𝑛 ∑ 𝑛 𝑖 ′
𝑖 =
𝑛 ′
𝑛 . (A.3)
First, by the definition of covariance, the covariance between growth rate 𝑤 𝑖 and
the trait value 𝑧 𝑖 is
cov(𝑤 𝑖 ,𝑧 𝑖 ) = E(𝑤 𝑖 𝑧 𝑖 )− E(𝑤 𝑖 )E(𝑧 𝑖 ) = E(𝑤 𝑖 𝑧 𝑖 )− 𝑤̅𝑧 ̅ , (A.4)
which is the numerator of the first term on the right-hand side of the Price equation.
Next, because ∆𝑧 𝑖 = 𝑧 𝑖 ′
− 𝑧 𝑖 and the expected value operator E(∙) is linear, the
numerator of the second term of the Price equation is
E(𝑤 𝑖 Δ𝑧 𝑖 ) = E[𝑤 𝑖 (𝑧 𝑖 ′
− 𝑧 𝑖 )] = E(𝑤 𝑖 𝑧 𝑖 ′
)− E(𝑤 𝑖 𝑧 𝑖 ) (A.5)
Combining equations (A.4) and (A.5) gives
215
cov(𝑤 𝑖 ,𝑧 𝑖 )+ E(𝑤 𝑖 Δ𝑧 𝑖 ) = E(𝑤 𝑖 𝑧 𝑖 )− 𝑤̅𝑧 ̅ + E(𝑤 𝑖 𝑧 𝑖 ′
)− E(𝑤 𝑖 𝑧 𝑖 )
= E(𝑤 𝑖 𝑧 𝑖 ′
)− 𝑤̅𝑧 ̅ (A.6)
The first term of the right-hand side of equation (A.6) can be simplified as
E(𝑤 𝑖 𝑧 𝑖 ′
) =
1
𝑛 ∑ 𝑤 𝑖 𝑧 𝑖 ′
𝑛 𝑖 𝑖
by the definition of
mean
=
1
𝑛 ∑
𝑛 𝑖 ′
𝑛 𝑖 𝑧 𝑖 ′
𝑛 𝑖 𝑖 =
1
𝑛 ∑ 𝑛 𝑖 ′
𝑧 𝑖 ′
𝑖 by (A.2)
= 𝑤̅
1
𝑛 ′
∑ 𝑛 𝑖 ′
𝑧 𝑖 ′
𝑖 by (A.3)
= 𝑤̅𝑧 ̅ ′ by the definition of mean
Hence,
cov(𝑤 𝑖 ,𝑧 𝑖 )+ E(𝑤 𝑖 Δ𝑧 𝑖 ) = 𝑤̅𝑧 ̅ ′− 𝑤̅𝑧 ̅
By dividing both sides by 𝑤̅, we obtain
cov(𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 Δ𝑧 𝑖 )
𝑤̅
= 𝑧 ̅
′
− 𝑧 ̅ = Δ𝑧 ̅
The right-hand side of this equation, 𝑧 ̅
′
− 𝑧 ̅ , is the change in the average trait of the
population from one generation to the next generation, Δ𝑧 ̅ .
A1.4. Simplification
When the trait 𝑧 𝑖 of group i does not change from the parent to the child
generations (i.e., with no mutation), that is, ∆𝑧 𝑖 = 𝑧 𝑖 ′
− 𝑧 𝑖 = 0, the second term in the
Price equation becomes zero resulting in a simplified version of the Price equation:
216
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 Δ𝑧 𝑖 )
𝑤̅
=
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
+
E(𝑤 𝑖 ∙ 0)
𝑤̅
=
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑧 𝑖 )
𝑤̅
.
(A.7)
Further, the growth rate itself is the trait of interest, that is, 𝑤 𝑖 = 𝑧 𝑖 .
Δ𝑧 ̅ =
𝑐𝑜𝑣 (𝑤 𝑖 ,𝑤 𝑖 )
𝑤̅
=
𝑣𝑎𝑟 (𝑤 𝑖 )
𝑤̅
.
(A.8)
The left-hand side represents “the rate of increase in fitness of any organism at any time,”
and the right-hand side represents “its genetic variance in fitness at that time.” Together,
it becomes Fisher’s (1930) fundamental theorem of natural selection: “The rate of
increase in fitness of any organism at any time is equal to its genetic variance in fitness at
that time” (p. 50).
Abstract (if available)
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