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USC Computer Science Technical Reports, no. 782 (2003)
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USC Computer Science Technical Reports, no. 782 (2003)
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Content
1
On Characterizing Network Hierarchy
H. Tangmunarunkit
USC-ISI
R. Govindan
USC
S. Jamin
Univ. of Michigan
S. Shenker
ICIR
W. Willinger
AT&T
Abstract— Our previous work in topology charac-
terization and hierarchy [1] introduced a hierarchy
metric to explore the hierarchical structure in vari-
ous networks. This metric is non-intuitive and com-
plicated. In this paper, we propose a simpler and more
natural metric for measuring network hierarchy. This
simpler metric uses slightly different criteria in select-
ing backbone links than the more complicated one.
Nevertheless, the network classifications according to
both metrics agree with each other. Furthermore, we
have extended the hierarchy analysis to examine path
characteristics and found that the hierarchical nature
of degree-based networks better resembles the hierar-
chy of the Internet at the AS level than at the router-
level.
I. INTRODUCTION
Modern network topology generators can be clas-
sified into two families—structural generators that
treat hierarchy as fundamental and degree-based
generators that treat the degree distribution as fun-
damental. Structural generators, such as tiers [2]
and transit-stub [3], create networks with a delib-
erately hierarchical structure. Degree-based genera-
tors, such as Brite [4] and Inet [5], on the other hand,
focus solely on generating networks with power-law
degree distributions to match the degree distribution
of the Internet [6]. Our previous work has shown that
degree-based generators that make no attempt to cre-
ate a hierarchical structure best model the large scale
structure of the Internet [1]. However, the Internet is
believed to have a significant degree of hierarchy; at
the router level, network engineers routinely speak of
backbones and at the AS level ISPs are broken into
different “tiers.”
Motivated by this seeming paradox, our previous
work introduced the hierarchy analysis using a com-
plicated metric, the minimum weighted vertex cover
(WVC), to measure the level of hierarchy in various
networks. Based on this metric, we classified net-
works into three different categories depending on
their level of hierarchy—strict, moderate or loose.
However, the WVC metric is non-intuitive and also
quite complicated. Therefore, in this paper, we pro-
pose a simpler and more natural metric—weighted
traversal set—for measuring network hierarchy. The
two metrics use different criteria for assigning value
to links and therefore select different sets of links as
backbone links. While WVC attempts to find the
minimum set of nodes that are affected by the re-
moval of the link, the simple metric focus entirely on
the the number of paths (between all end-points) go-
ing through the link. Despite their difference, the two
metrics’ classifications of networks into three differ-
ent groups agrees with each other. Therefore, the
result of this additonal metric reinforces the result of
our previous work.
This paper is written to complement our previous
work in topology characterization and hierarchy [1],
focusing entirely on hierarchy analysis. In addition
to the introduction of an additonal metric for measur-
ing hierarchy, we discuss issues that we experienced
researching for metrics to measure network hierar-
chy. Furthermore, we extend the analysis to inves-
tigate the charactertisic of the hierarchy by examin-
ing path characteristics. We found that, although the
two Internet representations (the autonomous system
connectivity map and the router connectivity map)
are classified to have the same moderate level of hi-
erarchy, their hierarchical characteristics are differ-
ent. Moreover, we found that the hierarchical na-
ture in degree-based networks better resembles the
AS hierarchy than the RL hierarchy. Similar to the
Tree network, the hierarchy in AS, degree-based,
and Transit-stub networks is strictly-structured while
very little structure is found in Tiers and RL map.
The paper is organized as follows. Section II
presents related work. Section III describes the two
metrics for computing the link usage value, includ-
ing the explantion of their difference. Section IV
presents the results of applying the simple metric to
2
various networks. We discuss the results and point
out issues that we encoutered searching for hierar-
chy metrics. Section V explores the characteristic of
network hierarchy. Section VI presents the hierarchy
analysis on other degree-based generators. Finally,
Section VII concludes our paper.
II. RELATED WORK
Although there is a large literature on routing hi-
erarchies, we are not aware of much work that has
attempted to measure (as opposed to create, or uti-
lize) hierarchy in network topologies. For example,
in the area and landmark hierarchy [7], the hierarchy
is created on top of the underlying topology. A few
nodes are recursively selected to represent a group
of nodes at various levels depending on the radius
of the area. Two recent and related examples [8],
[9] utilize the hierarchical property of AS paths to
infer business relationships (e.g., provider-customer)
in the AS topology. The latter work also classifies
ASs into a five-level hierarchy based on their busi-
ness relationships.
Probably closer in spirit to hierarchy measurement
is the classification of nodes into different levels of
hierarchy based on different criteria. Vukadinovic et
al. evaluate the Laplacian eigenvalue spectrum of a
variety of graphs and classify AS nodes into 5 dif-
ferent levels based on the node’s adjacency list [10].
Govindan et al. classify AS nodes into different
groups based on their outdegrees [11]. These ef-
forts attempt to characterize different nodes within
the same topology, instead of characterizing differ-
ent networks into different categories. Also closely
related to this work is our previous work in topology
characterization and hierarchy [1], [12]. As men-
tioned earlier, in [1], we use only the weighted ver-
tex cover metric, a complicated metric, for analyzing
network hierarchy, but without a simpler metric. We
did not examine path characteristics either.
We analyze various network generators in our
work. Network generators can be separated into two
different periods; the pre-1999 and post-1999 gener-
ators. Waxman [13], Transit-Stub [3] and Tiers [2]
were propsed before 1999. In 1999, Faloutsos et
al. reveals that the Internet’s degree distribution is
a power-law [6]. Because the pre-1999 network gen-
erators do not capture this phenomenon, many re-
searchers have proposed new generators that try to
capture the power-law degree distribution. Exam-
ples of these degree-based generators are found in
[5], [4], [14], [15], [16]. We describe some of these
generators in slightly more detail in Section VI.
Two representations of Internet topology, the AS
an RL maps, are also included in our analysis. The
RL map would not have been available for analysis
without developments in Internet router-level topol-
ogy discovery. Early work in this area used tracer-
outes from a small set of sources to several thousand
hosts to compute a router-level map [17]. Subse-
quent work improved the coverage of the Internet ad-
dress space by randomly selecting IP addresses [18],
randomly selecting addresses from route entries in
BGP tables [19], using a precomputed set of Web
sites [20], or using heuristics to infer addressable
parts of the IP space [21]. This last work also doc-
uments several techniques for improving complete-
ness of the inferred topologies.
III. HIERARCHY METRICS
Our notion of hierarchy revolves around the intu-
ition that there is a set of important links or back-
bone links that carry the traffic from many source-
destination pairs. We implicitly assume that the traf-
fic is not evenly spread out among the links but in-
stead is funneled into more central backbones. We
therefore conjecture that a symptom of hierarchical
structure is that some links are used more than oth-
ers. Here we are not referring to the level of traf-
fic as a function of the sending patterns of individual
hosts, but rather usage as measured by the set of node
pairs (source-destination pairs) whose traffic would
traverse the link; we call this the link’s traversal set.
Implicitly, in computing the traversal set for a
link l, we include all source-destination pairs whose
shortest paths include the link l. For the AS and RL
graphs, we extended this to account for policy rout-
ing. We use the policy model described in [22]. In
summary, at the AS level, this policy model com-
putes the shortest AS path between two nodes that
does not violate provider-customer relationships. An
example of a path that would violate these relation-
ships is one that traverses a provider, followed by a
customer and then back to another provider. We use
Gao’s technique [8] to infer peering relationships. To
compute the policy path in the RL graph, we first
compute the corresponding AS-level policy path, and
3
(u,v,l) w
(u,l) W
( , ,l) w a d
H
l
( ,l) W a
uv # paths through l
uv # paths
Link value of l
(u,l) W
H
l
(u) degree in
Link value of l
(u,v,l) w
(a,d),(a,e),(a,f),(a,g),
Traversal set for link l
(b,d),(b,e),(b,f),(b,g),
(c,d),(c,e),(c,f),(c,g)}
{
a
b
d c
e
f
g
l
G
d
e
f
g
c
b
a
=
=
Min weighted vertex cover
Vertex cover: C = {a,b,c}
= = 3
(u,v,l) w
H
l
v, (u,v)
u C
Weighted traversal set
= = 12
H
l
(u,v)
Fig. 1. An illustration of link value computation
then use shortest-paths within the sequence of ASs to
determine a router-level policy path.
We will now describe a simple metric—weighted
traversal set—for measuring the importance of each
link. The metric computes the link’s usage value
based entirely on the traffic which is measured based
on the number of end-points that go through the
link. For completeness, we will also summarize a
more complicated metric—weighted vertex cover—
introduced in [1] for comparison.
A. Weighted Traversal Set (WTSET)
The natural metric for measuring the hierarchy is
number of node-pairs that use a particular link to
carry its traffic, i.e., the size of the traversal set of a
particular link. However, for any source-destination
pair in the topology, there may be multiple equal-
cost paths connecting between them. Therefore, we
appropriatly assign the weight to each of the node
pair in the traversal set. For each node-pair (u; v )
in the traversal set of link l, we associate the weight
w (u; v ; l ) which is the fraction of the total number of
equal cost paths between u and v that traverse link l.
Thus, if there are multiple shortest paths between a
node pair, the contribution of the node pair is accord-
ingly weighted. We then modify our hierarchy met-
ric to measure the link’s value based on the weight
of each node pair in the traversal set instead of its
size. Specifically, for any link l, we define l’s value
to be the sum of w (u; v ; l ) where (u; v ) are included
in link l’s traversal set. See Figure 1 for an example.
B. Weighted Vertex Cover (WVC)
A more complicated way of computing a link’s
value is based on the vertex cover
1
of the link’s
traversal set. The vertex cover of a traversal set is
the minimum number of nodes that need to be re-
moved to eliminate at least one node from each pair
in the traversal set. For instance, a access link has
a vertex cover of 1, since eliminating the singleton
node eliminates all pairs from the set. Intuitively, the
vertex cover counts the smallest set of nodes affected
by removal of the link. A link for which this number
is high is more important (i.e., more nodes depend
on this link) than links for which the number is low.
To use this metric in the presence of multiple
shortest paths, we had to use a weighted vertex
cover
2
. Similar to the procedure in Section III-A,
a weight w (u; v ; l ) is assigned to each node-pair in
the traversal set of link l. Consider now the bipar-
tite graph (H
l
) formed by the traversal set (see Fig-
ure 1). Each vertex u in this graph is also assigned
a vertex weight W (u; l ) which is simply the average
w (u; v ; l ) such that (u; v ) belongs to the traversal set,
i.e., W (u; l ) is the average weight of the links that
are incident on node u in H
l
. We define a link’s
value to be the weight of the minimum weighted
vertex cover in the bipartite graph. A well-known
approximation algorithm [23] is used for computing
weighted vertex covers.
Example Figure 1 illustrates the link value com-
putation on a small graph. The topology of inter-
est is depicted by the graph G. Let us assume that
we want to compute the usage value of link l con-
necting between node c and d. The first step is to
compute the traversal set of link l. We then gener-
ate a bipartite graph H
l
according to the traversal
set. Each link connecting between node pair (u; v )
in the bipartite graph H
l
is then associated with the
weight w (u; v ; l ) which is the ratio of the number of
paths between u and v that traversed link l to the to-
tal number of paths between u and v. Since there
is only a single path connecting between any two
nodes, all the links weigh 1. After we are done as-
1
The formal defintion of a vertex cover problem is that for For
a given graph G(V; E ), we want to find C V such that for all
e = fu; vg2 E; e \ C 6=0 and C is the smallest set.
2
For a graph G(V; E ) and a positive weight function w : V !
R
+
on the vertices, we want to find C V such that for all
e = fu; vg2 E; e \ C 6=0 and w (C )
is minimum.
4
signing weight to each of the link in H
l
, we then
compute the weight W (u; l ) of each node u which is
the average link weight of u. In this case, the aver-
age link weight of all the nodes in H
l
is 1. Now we
are ready to compute the usage value of link l. Ac-
cording to the weighted traversal set, the link value
is the total weight of all the links in H
l
which is 12.
According to the weighted vertex cover metric, the
link value is minimum weight of the vertex cover set
which is 3 in this example.
Discussion
We now explain the subtle difference between the
two metrics by using an example. Consider the ex-
ample topology shown in Figure 2. The dummy
topology consists of three domains with 8, 9 and 83
nodes. The table beneath the topology shows the link
values according to the two metrics.
...
a1
a2
a0
...
b1 b2 b8
b0
...
c0
c1
c2
9 nodes
8 nodes
c7
83 nodes
a82
Link WTSET WVC
a0–b0 747 9
a0–c0 664 8
b0–c0 72 8
ai–bj, i=1,...,82;j=1,...,8 99 1
bi–cj, i=1,...,8; j=1,...,7 99 1
ai–cj, i=1,...,82;j=1,...,7 99 1
Fig. 2. Example topology
Based on the weighted traversal set metric, the ac-
cess links (such as c0-c1) are more important than
the inter-domain link b0–c0. This is because the
link b0–c0 is only used between the two small do-
mains and therefore does not have much traffic go-
ing through it. However, the weighted vertex cover
metric gives more value to the inter-domain link b0-
c0 than the access links because it is used to serve a
larger variety of nodes than the access links. For ex-
ample, if we remove one of the access links, say c0–
c1, c1 is the only one that loses connectivity to the
rest of the world. However, the disappearance of the
link b0–c0 affects all the nodes in the two domains;
they are forced to use longer paths to communicate
between them.
In summary, the weighted traversal set metric em-
phasizes the links with a large volume of traffic, i.e.,
the weight of the source-destination pairs that tra-
verse the link, independent of who the senders or re-
ceivers are. On the other hand, the weighted vertex
cover metric examines the list of node pairs associ-
ated with the link and emphasizes links that are used
by a variety of nodes. Another question, not of pri-
mary concern in this paper, is which set of these links
are considered more realistic backbones in the Inter-
net? To answer this question, we would need to have
network properties such as link bandwidth informa-
tion or node geographic location, none of which we
currently have. Therefore, we leave this question for
future work.
IV. RESULTS &DISCUSSION
We first describe the networks that we included
in our analysis. These networks are the same set of
networks used in our earlier work [1].
A. Networks
We include three categories of network graphs:
measured networks, generated networks, or canoni-
cal networks. We summarize these categories below.
1) Measured Networks: Two representations of
the Internet topology are included. The first one is
the AS topology, representing inter-autonomous sys-
tem (AS) connectivity. Nodes in this topology rep-
resent ASs, and links represent peering relationships
between them. The second measured topology is the
Internet router-level (RL) topology. Nodes in this
topology represent routers, and links connect routers
that are one IP-level hop from each other. The detail
about how the maps are collected and the caveat of
map collections are reported in [1], [21].
2) Generators: Artificial networks generated
from three families of generators are used in the
study.
Random Graph Based Generators. This class
is represented by the Waxman [13] generator
which is a variant of the classical Erdos-Renyi
random graph model [24].
Structural Generators. This class contains the
Transit-Stub [3] and Tiers [2] generators. Both
5
Type Topology Number of Nodes Avg. Degree Comment
Measured RL 170589 2.53 May 2001
AS 10941 4.13 May 2001
Generated Transit-Stub (TS) 1008 2.78 3006 0.55 6 0.32 9 0.248
Tiers 5000 2.83 15010500 40520201201
Waxman 5000 7.22 5000 0.005 0.30
PLRG 9230 4.46 2.246
Canonical Tree 1093 2.00 k=3,D=6
Mesh 900 3.87 30x30 grid
Random 5018 4.18 Link prob = 0.0008
Fig. 3. Table of network topologies used. See [12] for a description of parameters for the generated networks.
generators attempt to mimic the hierarchical
structure of the Internet by explicitly construct-
ing networks with hierarchy. Transit-Stub em-
beds the notion of transit and stub domains
inside the networks. Tiers embeds a notion
of Wide Area Network (WAN), Metropolitan
Area Network (MAN), and Local Area Net-
work (LAN).
Degree-based Generators. This class is rep-
resented by the simplest degree-based gen-
erator, called the power-law random graph
(PLRG) [16]. For a given a target number of
nodes N, and an exponent , PLRG first as-
signs degrees to N nodes drawn from a power-
law distribution with exponent (i.e., the prob-
ability of a degree of k is proportional to k
).
Two nodes are then selected to connected to-
gether, each of which is selected with the prob-
ability that is proportional its number of un-
matched links. The number of unmatched links
of each node, first assigned by its degree, is
decremented by one every time it is selected.
The analysis of other degree-based generators
is included in Section VI.
3) Canonical Networks: Finally, the third group
of networks contain the k-ary Tree, the rectangular
grid or Mesh, and an Erdos-Renyi Random graph.
These networks are included to calibrate our results.
B. Results
We now describe the results of applying the
weighted traversal set metric to compute link values
of specific instances of networks presented in Fig-
ure 3.
If a network is hierarchical, we would expect dif-
ferent sets of links to have different link values, e.g.,
the backbone links should have much higher values
than peripheral links. On the other hand, if all links
have similar values (indicating the same degree of
importance), then the network has loose or no hier-
archy. Therefore, similar to our previous work, we
use the distribution of link values and the magnitude
of highest link values as our measurements of net-
work hierarchy.
Figures 4(a)-(c) show the link value distributions
for the canonical, generated, and measured networks.
In these plots, the x-axis plots the rank of a link ac-
cording to its value (a higher rank indicating a higher
value), normalized by the number of links in the
topology. The y-axis depicts the link value normal-
ized by the n
2
where n is the number of nodes in the
network; the largest traversal set size is in the order
of O(n
2
). Figures 4(d)-(e) plot the same data but on
different scale (x-axis is on normal scale and y-axis
is on the log-scale). For completeness, we include
the link value distribution based on the weighted ver-
tex cover (Figure 13) in the appendix.
By examining Figure 4, we conclude that there ex-
ist three classes of hierarchy in our graphs: strict,
moderate, and loose. Figure 4(a)–(c) emphasize the
distribution of the high valued links in each net-
work. In terms of the magnitude of link values, the
data reveals that the highest link values in Tree, TS,
and Tiers are significantly higher than all the other
topologies; they are at least five times higher. In ad-
dition, their link value distributions fall off rapidly
suggesting that within the same topology, there is a
small set of backbone links that have much higher
values than the rest of the links. We say, by this mea-
sure, that these topologies have a strict hierarchy.
By examining the range of the distribution in Fig-
ure 4(d)–(e), we can further classify the remaining
networks into two other groupings. Even though the
magnitude of the highest value links in these net-
works are comparable, it is very obvious that the link
value distribution of Mesh, Random and Waxman
6
0
0.05
0.1
0.15
0.2
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
Tree
Mesh
Random
(a) Canonical
0
0.05
0.1
0.15
0.2
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
1
2 3
4
1 : AS(Policy)
2 : AS
3 : RL(Policy)
4 : RL
(b) Measured
0
0.05
0.1
0.15
0.2
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
TS
Tiers
PLRG
WM
(c) Generated
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Traversal Set
Normalized Link Rank
Tree
Mesh
Random
(d) Canonical
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Traversal Set
Normalized Link Rank
AS(Policy)
AS
RL(Policy) RL
(e) Measured
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Traversal Set
Normalized Link Rank
TS
Tiers
PLRG
WM
(f) Generated
Fig. 4. The link value rank distribution based on the weighted traversal set metric
graphs are very flat, e.g., their ranges of link values
only span one or two orders of magnitude. This indi-
cates that most of the links within the same network
have similar values. Therefore, we classify them into
a group of networks that have loose or no hierarchy.
This is consistent with generally accepted wisdom
about the lack of hierarchy in the mesh and the ran-
dom graph.
Measured networks and PLRG, on the other hand,
have a wider range of values than that of Mesh, Ran-
dom and Waxman; they span 4 to 5 orders of mag-
nitude. Moreover, similar to strict hierarchical net-
works, we also observed the rapid fall-offs in these
networks. For example, the first 10% of the links in
the AS graph falls off from 0.01 to 0.0001. There-
fore, we describe AS, RL
3
and PLRG as having a
3
Computing the weight associated with each link in the bi-
partite graph of the full RL graph is computationally expen-
sive. Therefore, we compute the link values of the core topology
(generated by recursively removing degree 1 nodes) instead. In
previous work, we have found that link values (computed in a
slightly different way) computed on the core map correlate well
with link values obtained from a full map.
moderate hierarchy. In fact, the other degree-based
generators that we evaluate in section VI also fall
into this category.
The table below depicts these qualitative group-
ings.
Topology Strict Moderate Loose
Mesh x
Random x
Tree x
AS, RL, PLRG x
Tiers x
TS x
Waxman x
Consistent with the results of weighted vertex
cover, we found that:
Accounting for policy routing in computing
the link values does not qualitatively alter our
groupings. As expected with policy routing,
since paths are more concentrated, the highest
link values are larger than with shortest path
routing both for AS and RL.
The structural generators construct a much
stricter form of hierarchy than is present in the
7
measured graphs.
PLRG qualitatively models the hierarchy
present in AS and RL graphs, even after accou-
ting for policy routing.
C. Validation
As mentioned earlier, the backbone links are ex-
pected to have higher values than peripheral links.
We have verified, for several of our topologies, that
this expectation holds. In the tree topology, the high-
est valued links are located near the root of the tree.
As we traverse from the root to any leaf, the link
values get smaller. In TS, the highest valued links
are located in the transit clouds. In Tiers they are
in the WAN. In the AS graph, high value links are
those that connect well-known national backbones
together. Figure 5 show the top ten highest value
links in the AS topology. Finally, in the RL graph
they occur in, or between, these backbone ASs, as
shown in Figure IV-C. This provided a sanity check
on our metric.
Alternet : UUNET
Alternet : SprintLink
CW : Alternet
Alternet : AT&T
Rostelecom : CW
Nacamar : Alternet
Globix : Alternet
Alternet : UUNET
Abovenet : Alternet
Teleglobe : Alternet
Alternet : UUNET
SprintLink : Alternet
CW : Alternet
Alternet : AT&T
SprintLink : CW
CW : Rostelecom
CW : AT&T
Ebone : Alternet
SprintLink : Teleglobe
Alternet : Teleglobe
(a) Shortest Path Routing (b) Policy Routing
Fig. 5. Top 10 highest links on the AS map based on WTSET
Alternet : Alternet
CW : AT&T
Alternet : TELSTRA-AS
Alternet : Alternet
SprintLink : SprintLink
Qwest : KDDI
SprintLink : SprintLink
CW : CW
Qwest : Teleglobe
CW : CW
Alternet : Alternet
Alternet : Alternet
Alternet : Alternet
Alternet : Alternet
Alternet : Alternet
SprintLink : SprintLink
Alternet : Alternet
Alternet : Alternet
SprintLink : SprintLink
SprintLink : SprintLink
(a) Shortest Path Routing (b) Policy Routing
Fig. 6. Top 10 highest links on the RL map based on WTSET
We have done the same validation on the weighted
vertex cover metric and found similar results. The
top ten highest value links of the AS (Figure 14) and
RL (Figure 15) graphs according the weighted vertex
are included in the appendix.
D. Discussion
This section discusses the results of the two met-
rics and the necessity of appropriately accounting for
multiple paths in the hierarchy metrics.
1) Weighted Traversal Set versus Weighted Vertex
Cover: Despite the difference between the two met-
rics, we found that both of them satisfy our valida-
tion tests. Moreover, the qualitative network classi-
fication according to both metrics are also the same.
This motivates us to look at the correlation between
the two metrics.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mesh
TS
Tiers
Random
Waxman
RL(Policy)
RL
AS(Policy)
AS
PLRG
Correlation
Tree
Fig. 7. Correlation between the two metrics
Figure 7 shows the correlation between the two
hierarchy metrics. We found relatively high correla-
tion between the two metrics on networks in the strict
and loose group; e.g., links that have high/low value
based on the WTSET metric tends to have high/low
value based on the WVC metric. Networks in the
moderate hierarchical group, especially PLRG, seem
to have relatively lower correlation. For these net-
works, the two metrics probably select different sets
of links as backbone links.
4
However, since the mea-
sure of network hierarchy is based on the macro-
scopic property such as the distribution of the link
values, the difference between the two metrics does
not affect our qualitative classification. Therefore,
the results in this paper reinforce the previous results
in [1].
4
We verify this by examining the cardinality of the intersec-
tion set of links that are in the top 1%, 5% and 10% according to
both metrics. The cardinality of these intersection sets are about
50-60% for measured networks and 20% for PLRG. The car-
dinality of other networks are relatively higher; they are in the
range of 70% to 100%. This validates that the backbone links
according to the two metrics are relatively different.
8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
Tree
Mesh
Random
(a) Canonical
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
RL
AS
AS(Policy)
RL(Policy)
(b) Measured
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Traversal Set
Normalized Link Rank
TS
Tiers
PLRG
WM
(c) Generated
Fig. 8. The link value rank distribution based on the (unweighted) traversal set size metric (x-axis on log scale)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-06 1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
Tree
Mesh
Random
(a) Canonical
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-06 1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
RL(Policy)
RL
AS
AS(Policy)
(b) Measured
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-06 1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
TS
Tiers
PLRG
WM
(c) Generated
Fig. 9. The link value rank distribution based on the (unweighed) vertex cover metric (x-axis on log scale)
2) Unweighted versus Weighted Metrics: Prop-
erly accounting for multiple paths between any
source-destination pair turns out to be essential in
our study. Without appropriate weight, these two
simple measures are misleading. We explain the
flaws of the unweighted version of the two metrics
below.
a) Traversal Set versus Weighted Traveral Set:
Figure 8 shows the distribution of traversal set sizes
(with no accounting for multiple paths). This met-
ric does not discriminate the Mesh from other strict
hierarchical networks; in fact the highest value of
the Mesh is higher than other strict hierarchical net-
works. The number of paths between any two nodes
in the Mesh is quite large; the further away the end-
nodes are, the larger the number of paths between
them. Without appropriately weighting the contribu-
tion of each node-pair on the link, we observe large
number of pairs, e.g., large traversal set size, espe-
cially at the center of the Mesh. Moreover, according
to this metric, the two measured networks with short-
est path routing shows higher level of hierarchcy
comparing to those with policy routing; this result is
counter-intuitive. Policy routing funnels many paths
of some source-destination pairs through particular
parts of the networks, resulting in larger traversal set
sizes for backbone links. However, policy routing
also restricts possible paths between any two nodes,
resulting in the reduction of the traversal set size of
some links. Figure 8 shows that, without appropri-
ately weighting the contribution of each node pair,
the traffic reduction effect is more dominant than the
traffic increase due to policy routing. This result in-
dicates that policy routing reduces the level of hier-
archy. This is inaccurate, so we disregard this metric
in the study.
b) Vertex Cover versus Weighted Vertex Cover:
Figure 9 shows the distribution of the vertex
cover size. According to this metric, each source-
destination pair in the traversal set is assigned by
9
equal weight. Therefore, for each link, instead of
computing the weight of the vertex cover set that
yields minimum weight, we compute the size of the
vertex cover. The flaw of this metric is similar to that
of the (unweighted) traversal set. The metric does
not discriminate the Mesh from other strict hierar-
chical networks, and it indicates that the policy rout-
ing reduces the level of hierarchy in the AS network.
Thus, we also disregard this metric in our study.
V. HIERARCHICAL CHARACTERISTIC
What is the hierarchical characteristic of these net-
works? To answer this question, we investigate the
path charactersistic between any node-pairs in the
network. For any source-destination pair, we exam-
ine the sequence of link values along the path. In-
tuitively, if a network has a strictly-structured hier-
archy, such as in a tree topology, we would expect
these sequences to be rising as we traverse from a
leaf to the root of the tree, falling as we traverse
from the root of the tree to any leaf, or rising and
then falling as we traverse between any leaf or inter-
nal nodes. Based on this intuition, we classify a path
into one of the two categories.
valid paths: A path is valid if the sequence of
link values are flat, rising, falling, or rising and
then falling.
invalid paths: Any path that is not valid is clas-
sified as an invalid path. An example of an in-
valid path is the path that has values falling and
then rising.
1
0.8
0.6
0.4
0.2
0
Tree
Mesh
TS
AS(Policy)
AS
PLRG
Waxman
Random
RL
Tiers
RL(Policy)
Fraction of valid paths
WTSET
WVC
Fig. 10. Fraction of valid paths
Figure 10 shows the fraction of paths that are valid
based on the two metrics. As expected, all paths
on the Tree are valid. TS, AS and PLRG have rel-
atively high fraction of paths that are valid. This re-
sult indicates that the hierarchy in these networks are
quite strictly-structured and the arrangement of high
value links is approximately tree-like. To the con-
trary, Tiers and RL have very low fractions of valid
paths. This indicates that the hierarchy in these net-
works are not strictly-structured. High value links in
these networks are distributed across the network.
In summary, we found that:
the hierarchical structure of the AS and RL map
are quite different even though both of them are
classified as having moderate level of hierarchy,
and
PLRG (along with other degree-based net-
works; Section VI) resembles the hierarchical
nature of the AS beter than the RL map.
Note that the fraction of valid paths in the RL map
with policy routing is less than that with the shortest
path routing. This is because there might not exist
any path between some source-destination pairs in
the network. This is due to the incomplete of the
graph and the error of the heurististic in identifying
the peering relationship on the AS map [21], [25].
VI. OTHER POWER-LAW VARIANT
GENERATORS?
We explore other degree-based generators (gener-
ators that generate power-law degree distributions)
in this section. In previous sections, we have used a
single degree-based generator, the PLRG. The PLRG
generator uses a particularly simple technique for
connecting nodes (Section IV-A). It clones each
node as many times as the degree assigned to it, then
uniformly randomly connects the clones. However,
given a set of nodes with a particular degree distri-
bution (such as a power-law distribution), nodes can
be connected in different ways to satisfy the degree
requirements.
One class of approaches to node connectivity is
exemplified by the model proposed by Barabasi and
Albert [26]—we call this the BA model—and the
Brite generator. The BA model is an evolutionary
process that generates graphs with power-law de-
gree distributions. The graph is grown incrementally,
with newly appearing nodes randomly connecting to
10
already existing nodes, but in proportion to their de-
grees. The Brite [4] generator incorporates the BA
model with additonal features, such as node place-
ment (random or heavy-tail) and geographic bias in
establishing links. We used a heavy-tailed option
when generating a network in our study. However,
we did not explore the latter feature. A slight vari-
ant of the BA model proposed by the same authors
incorporates link addition and re-wiring [14]; with a
small, but uniform probability a link can be added
between two nodes, or an existing link can reattach
from one endpoint to another based on preferential
connectivity. Later, this variant has been modified
by Bu and Towley—we call the modified version the
BT model—to allow more flexibility in specifying
how the nodes are connected.
Another class of approaches initially assigns node
degrees from a power-law degree distribution, simi-
lar to the PLRG. Unlike the PLRG, however, these
approaches connect nodes using different rules. For
example, after conducting a feasibility test on the
generated degree distribution to see if the resulting
graph would be connected, the Inet [5] generator cre-
ates a spanning tree among nodes of degree larger
than one, connects degree one nodes to this span-
ning tree with proportional connectivity,
5
then sat-
isfies the degrees of remaining nodes in decreasing
degree order. Another generator [15] connects the
nodes randomly, without cloning.
A. Results
Figure 11 and Figure 12 show the link value dis-
tributions for the PLRG-variant networks and mea-
sured networks based on the weighted traversal set
and weighted vertex cover respectively.
6
Similar
to the measured networks, the distributions of the
PLRG-variants networks falls off quickly. The high-
est value links are approximately in the same range
as those of measured networks with BT as an excep-
tion based on the weighted traversal set.
The highest value of link of the BT network
is slightly lower than other degree-based networks.
5
The likelihood of attaching to a node is proportional to its
degree
6
Our previous paper [1] mentioned briefly in words that other
degree-based generators are also classified as having moderate
hierarchy according to the weighted vertex cover. For clarity
and completeness, we include the distribution of degree-based
networks according the two models in this paper.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
AS(Policy)
AS
RL
RL(Policy)
(a) Measured Networks
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
BA
Inet
Brite
BT
PLRG
(b) PLRG Variants
Fig. 11. The link value rank distribution based on WTSET
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
RL(Policy)
RL
AS(Policy)
AS
(a) Measured Networks
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
Normalized Link Rank
BA
Brite
BT
Inet
PLRG
(b) PLRG Variants
Fig. 12. The link value rank distribution based on WVC
This is because the average degree of BT is twice
as large as those of other degree based; the aver-
age degree of the BT is about 9 while other net-
works have the average degree of 4. Therefore, there
will be many more paths connecting between any
two nodes and hence reducing the weight of each
source-destination pair in the traversal sets, which
as a results lower the value of the links in the net-
work. However, similar to some other degree-based
networks, the range of link values in the BT networks
is wider comparing to that of the Random graph or
Mesh. Therefore, we conclude that, as the AS and
RL networks, the PLRG-variant networks can be
described as having a moderate hierarchy. Further
more, the fraction of valid paths of these networks
are also in the same range as that of AS, which is
much more than that observed in the RL map. There-
fore, we conclude that the nature of these networks
better resembles that of the AS, than the RL map.
11
VII. CONCLUSION &FUTURE WORK
Our earlier work in topology characterization and
hierarchy [1] has introduced a hierarchy metric for
measuring network hierarchy. This metric is non-
intuitive and complicated. In this paper, we proposed
an additional metric which is simpler and more nat-
ural. The two metrics use different criteria for as-
signing a link’s usage value and therefore select dif-
ferent sets of links as backbone links. The differ-
ence between the two metrics was discussed in Sec-
tion III and IV-D. However, despite their difference,
we found that the classification of various networks
agree with each other. Furthermore, we have also
examined path characteristics which reveals that the
hierarchical structure of degree-based networks is
more similar to that of the AS than the RL network.
The natural follow-on question is which of these
sets of backbone links—WTSET-based or WVC-
based, are more realistic backbone links? We leave
this question for future work. Also, as an initial step
to understand the characteristics of the Internet struc-
ture, we have treated the AS and RL maps separately.
However, since both AS and RL maps reflect the
overall structure of the Internet, and the AS topol-
ogy is a logical overlay on the router-level topology,
one would expect that there is some hierarchical re-
lationship between the two maps. But what is this
relationship and how can we find it? We leave this
investigation for future work as well.
APPENDIX
WEIGHTED VERTEX COVER:LINK VALUE
DISTRIBUTION
Figures 13(a)-(c) show this distribution for the
canonical, generated, and measured networks. In
these plots, the x-axis plots the rank of a link ac-
cording to its value normalized by the number of
links in the topology. The y-axis depicts the link
value normalized by the number of nodes in the net-
work; the largest vertex cover set is in the order of
O(n) where n is the number of nodes in the network.
Figures 13(d)-(e) plot the same data but on different
scale.
VALIDATION OF WEIGHTED TRAVERSAL SET
Figure 14 and 15 show the top ten highest value
links with the shortest path and policy routing based
CW : SprintLink
Teleglobe : AT&T
Alternet : SprintLink
GTE : AT&T
Level 3 : UUNET
Teleglobe : Qwest
Alternet : UUNET
CW : AT&T
Abovenet : Qwest
CW : Alternet
CW : SprintLink
Teleglobe : AT&T
Alternet : SprintLink
GTE : AT&T
Level 3 : UUNET
Teleglobe : Qwest
Alternet : UUNET
CW : AT&T
Abovenet : Qwest
CW : Alternet
(a) Shortest Path Routing (b) Policy Routing
Fig. 14. Top 10 highest links on the AS map based on WVC
Alternet : Alternet
SprintLink : SprintLink
Alternet : TELSTRA
SprintLink : SprintLink
Alternet : Alternet
Qwest : Teleglobe
SprintLink : Teleglobe
Alternet : TELSTRA
CW : CW
CW : CW
Alternet : Alternet
Alternet : Alternet
Alternet : Alternet
SprintLink : SprintLink
Alternet : Alternet
SprintLink : SprintLink
SprintLink : SprintLink
SprintLink : SprintLink
CW : CW
SprintLink : SprintLink
(a) Shortest Path Routing (b) Policy Routing
Fig. 15. Top 10 highest links on the RL map based on WVC
on the WVC metric. Similar to WTSET metric, these
links connect big ISPs together in the AS map, and
in the RL map, they are either internal links of big
ISPs or interconnected link between them.
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0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1e-05 0.0001 0.001 0.01 0.1 1
Normalized Link Value
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(c) Generated
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Abstract (if available)
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Description
Hongsuda Tangmunarunkit, Ramesh Govindan, Sugih Jamin, Scott Shenker, Walter Willinger. "On characterizing network hierarchy." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 782 (2003).
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USC Computer Science Technical Reports, no. 782 (2003)
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