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USC Computer Science Technical Reports, no. 822 (2004)
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Content
An Empirical Evaluation of Internet Latency
Expansion
Hui Zhang Ashish Goel Ramesh Govindan
Abstract—
The Internet’s latency expansion determines the asymp-
totic performance of large-scale distributed systems (such
as Peer-to-Peer systems), but previous studies on the
Internet have defined expansion in terms of router-level
hops. In this paper, we empirically determine the Internet’s
latency expansion characteristics using measurements from
two different Internet topology datasets. Our results show
that the Internet router-level topology exhibits a power-law
latency expansion, in contrast to its exponential expansion
rate in terms of hops.
I. INTRODUCTION
The latency expansion rate is a basic characteristic of
a data communication network. With the recent research
focus on large-scale distributed systems (such as peer-to-
peer systems), the Internet’s latency expansion has been
shown to be an important determinant of asymptotic per-
formance of these systems. In this paper, we empirically
determine the Internet’s latency expansion characteristics
using measurements from two different Internet topology
datasets.
Formally, latency expansion describes the distribution
of the latency between node pairs in a network. Let G be
an undirected network G = (V; E), with N =jVj nodes
and a latency function l defined on the set of links. Let
N
u
(x) denote the set of all nodes v 6= u; v 2 V such
that l(u; v) x. A family of networks has a power-law
latency expansion if there exist constants c
1
; c
2
> 0 and
d
1
; d
2
1 such that for all networks G = (V; E) in the
family, and each node u2 V , the following are true:
1) jN
u
(x)j c
1
x
d1
for all x 0, and
2) jN
u
(x)j c
2
x
d2
for all x, where 1 x ((jVj
1))=c
2
)
1=d2
.
Hui Zhang is at the Department of Computer Science, University
of Southern California. This research was supported in part by NSF
grant CCR0126347. Email: huizhang@ usc.edu.
Ashish Goel is at the Departments of Management Science and
Engineering and (by courtesy) Computer Science, Stanford University.
This research was done while the author was at the University of South-
ern California, and was supported in part by NSF grant CCR0126347
and NSF Career grant No. 0133968. Email: ashishg@stanford.edu.
Ramesh Govindan is at the Department of Computer Science,
University of Southern California. This research was supported in part
by NSF grant CCR0126347. Email: ramesh@usc.edu.
Intuitively, in networks with power-law latency expan-
sion, the average number of nodes within distance x
of any node grows algebraically with x. Some simple
examples of network families with power-law latency
expansion are rings, lines, (both have d
1
= d
2
= 1), and
meshes (i.e., grids, d
1
= d
2
= 2), where each edge has
latency 1.
Analogously, we define the notion of exponential
latency expansion for a family of networks if there exist
constants c
1
; c
2
> 0 and
1
;
2
> 1 such that for all
networks G = (V; E) in the family, and each node
u2 V , the following are true:
1) jN
u
(x)j c
1
x
1
for all x, and
2) jN
u
(x)j c
2
x
2
for all x, where 1 x
log
2
((jVj 1)=c
2
).
Intuitively, in networks with exponential latency expan-
sion, the average number of nodes within distance x of
any node grows exponentially with x. Two families of
networks known to have exponential expansion are the
classical random graph [3] and the power-law random
graph (PLRG) [2], where each edge has latency 1.
The performance of many networking algorithms re-
lies on the latency expansion characteristic of the under-
lying network. For example, in early seminal work on
the problem of reducing request latency in web cache
systems, Plaxton, Rajaraman, and Richa [18] showed
that with carefully pre-configured routing tables, a data
replication system can achieve asymptotically optimum
request latency if the underlying network has power-
law latency expansion. Karger and Ruhl [7] designed
an elegant data structure for the nearest neighbor prob-
lem to achieve optimal performance in networks with
power-law latency expansion. Zhang, Goel, and Govin-
dan [23] showed that for a class of Distributed Hash
Table schemes, if the underlying network topology has
exponential latency expansion, then any optimizations
performed on them can only yield a constant factor im-
provement on the routing latency, which keeps increasing
with the overlay network size. However, the situation is
quite different for networks with a power-law latency
expansion: a simple random sampling scheme results
in near-optimal routing latency, which remains within
ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 93
a small constant factor of the underlying unicast latency.
Abraham, Mahlia, and Dobzinski [1] also showed that
their design of locality aware Distributed Hash Table
networks achieved constant lookup latency distortion un-
der power-law latency expansion metric spaces. Kempe
and Kleinberg [8] investigated gossip-based communi-
cation mechanisms, and they too obtained particularly
good results for metric spaces with power-law latency
expansion. As all the above pieces of work consider
the Internet as a major application, it is important to
understand the latency expansion characteristics of the
Internet.
The literature on topology modeling and characteri-
zation has generally assumed that real Internet router-
level topologies have an exponential expansion
1
[16],
[22]. However, previous studies have defined expansion
in terms of router-level hops. It remains to answer the
following question: what are the expansion character-
istics of the Internet router-level topology in terms of
latency?
This note presents an empirical effort to understand
the latency expansion in the Internet, and our prelimi-
nary results show that the Internet resembles power-law
latency expansion on the router-level.
II. METHODOLOGY
To empirically measure the latency expansion rate of
the Internet, we need the network distance information
between Internet nodes. Clearly, it is prohibitively ex-
pensive to measure pairwise latencies between all pairs
of nodes due to the size of the Internet. One possible
measurement technique uses a geometric embedding
approach proposed for network distance estimation [10],
[12], [20]. Even with such techniques, we believe that
a significant number of landmarks will be necessary to
get acceptably low error rates, and this approach presents
formidable logistical challenges.
Instead, we measure the latency expansion of two
router-level topologies gathered by mapping the Inter-
net [6]. The router-level topology datasets were gathered
using a methodology similar to prior work on topology
discovery: traceroutes from several locations (in this
case, six nodes on the NIMI infrastructure [14] for the
first topology dataset, and more than fifty nodes on the
PlanetLab platform [17] for the second topology dataset.)
to random destinations chosen from BGP routing tables.
Router interfaces were disambiguated using techniques
described in [6].
How do we measure latency on an Internet router-level
topology? We argue that, since the interested networking
1
Earlier work, based on a smaller dataset had claimed that the
expansion of router-level topologies better matched a power-law [5],
but the current belief is that this expansion is exponential.
algorithms mentioned in Section. I all use small-sized
messages in their overlay routing, the propagation la-
tency dominates the one-way latency between two nodes.
The expansion characteristic of the Internet router-level
topology that we are interested in, then, is the propaga-
tion latency expansion, which reflects the static property
of the network distance, rather than prevailing traffic
conditions.
To measure the propagation latency expansion of the
Internet router-level topologies, we make the following
simplifying assumption: the propagation latency between
any two nodes on a router-level topology is well approx-
imated by the cumulative geographic distance of the path
between the two nodes in shortest path routing. While
prior research [4], [15], [13], [21] has clearly pointed
out that the direct geographic distance between two
network nodes does not necessarily correlate well with
their propagation latency, we used the cumulative geo-
graphical distance of the routing path to more carefully
approximate the propagation latency. The one exception
to this is the case when a link in the router-level topology
corresponds to a link-layer tunnel or circuit that is
actually routed through one or more geographic locations
before reaching the end of the link. At the very least,
this geographic distance can be used to establish a lower
bound on the propagation latency for the given routing
path.
To assign geographic location to nodes in our router-
level topologies, we used the Geotrack tool [13]. This
tool uses heuristics based on DNS names to infer node
locations. For both data sets about 15% of all nodes
were assigned locations by Geotrack. While this num-
ber is not high, we believe it to be acceptable since
Geotrack’s heuristics identify the location of backbone
routers of major ISPs by using their DNS names. Our
work approximates the location of other nodes by the
topologically nearest ISP backbone router (if there was
more than one such neighbor, we randomly picked one).
This is reasonable, to a first order.
We collected the two Internet topology datasets inde-
pendently, one in May 2002 with 328378 routers, and the
other in November 2003 with 356645 routers. We call
the first topology T1 and the second one T2 in the rest of
the paper. The routers in T1 are mapped into 602 distinct
cities, and the routers in T2 are mapped into 613 distinct
cities. Figure 1 shows the router distribution among cities
for the node assignment used in the experiment, and
Table I presents a partial city list ranked by the number
of assigned routers.
Having assigned geo-locations to all nodes, we could
compute the latency expansion properties of the Inter-
net router-level graph. However, each of our router-
level topologies contained more than 300,000 nodes.
ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 94
Rank City in T1 City in T2
1 New York, United States New York, United States
2 Atlanta, United States Chicago, United States
3 Los Angeles, United States Washington, United States
4 Dalllas, United States Dallas, United States
5 Chicago, United States Frankfurt, Germany
6 Washington, United States Los Angeles, United States
7 London, United Kingdom Tokyo, Japan
8 San Francisco, United States London, United Kingdom
9 Frankfurt, Germany Atlanta, United States
10 Tokyo, Japan Palo Alto, United States
TABLE I
THE TOP-10 CITY LISTS IN THE TWO TOPOLOGIES RANKED BY THE NUMBER OF ASSOCIATED ROUTERS.
0
0.01
0.02
0.03
0.04
0.05
0.06
0 100 200 300 400 500 600 700
fraction of assigned routers
city ID
router distribution among cities
T1
T2
T1
T2
Fig. 1. router distribution in cities
To tractably compute the latency expansion of these
rather large topologies, we randomly sampled about
100,000 node pairs from each topology and computed
the geographic distance between them respectively.
2
Before we present our results, we mention that our
procedure is approximate for several reasons. The in-
completeness of topology discovery and interface dis-
ambiguation (or alias resolution) methods is well doc-
umented [6], [9], [19]. Geo-location techniques such
as Geotrack are also approximate; in particular, our
approximation of placing un-resolvable nodes near re-
solvable ones can underestimate the actual geographic
distance between two nodes. Despite this, we believe our
general conclusions from this experiment will hold up to
scrutiny, since we have been careful to make qualitative
judgments from a very large dataset.
2
We also attempted other sampling techniques: one-to-all distance
measurement for up to 1000 nodes, and full connection distance
measurement for large subgraphs up to 6400 nodes. All the expansion
curves converged to one another as the number of sampled nodes
increased.
III. RESULTS
Figure 2 plots the latency expansion of the router-
level topology in T2. It includes, for calibration, the
hop-count expansions of the ring, mesh and power-
law random graph (PLRG)
3
[2] topologies, as well as
the hop-count expansion of the router-level topology
itself. Besides, the latency expansion curves for both
router-level topologies are put together for comparison
in Figure 3. It’s well known that both a ring and a mesh
have power-law hop-count expansion while a PLRG
graph has exponential hop-count expansion. We imme-
diately make the following observations. The hop-count
expansion of the Internet router-level topology resembles
that of the PLRG and can be said to be exponential.
However, the latency expansion of the Internet router-
level topologies is significantly more gradual than its
hop-count counterpart and more closely matches the ring
and the mesh topologies.
To understand the latency expansion better, we fitted
the Internet latency expansion to several analytic curves.
Our fit focuses on the part of the empirical distribution
before it saturates. Note that in a finite graph, the latency
expansion will always slow down after some distance. In
our case, the threshhold distance is around 10000, be-
yond which less than 10% of node-pair distance samples
fall into. For the first 90% part (the value of y in [0;0:9])
of the Internet latency expansion curve, the linear curve
y = A + BX (A = 0:93; B = 8:23e 5) fits both
datasets well, as shown in Figure 4 for T1. We believe,
then, that the Internet has a power-law latency expansion
rate
4
.
3
Recent work [22] has shown that the PLRG is a reasonable model
for the Internet router-level topology, particularly when attempting to
evaluate large-scale metrics.
4
The experiments in [23] reaffirmed this conclusion: on the Internet
router-level topologies used in this paper, the topology-dependent
random sampling scheme performed qualitatively the same as it did
on a ring or mesh topology; the performance was totally different on
a PLRG graph.
ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 95
0.0001
0.001
0.01
0.1
1
10
100
0.0001 0.001 0.01 0.1 1
Probability[ distance < X ] (in log)
Distance X (in log and normalized by network diameter)
Expansion rates in different networks
T2 (Geographical Distance)
Ring
Mesh
T2 (Hop Distance) PLRG
Ring - 1000 nodes
Mesh - 1600 nodes
PLRG - 4232 nodes
T2 (Geographical Distance) - 112967 node-pair samples
T2 (Hop Distance) - 112967 node-pair samples
Fig. 2. Internet latency expansion (in log-log)
1e-05
0.0001
0.001
0.01
0.1
1
10
100
0.0001 0.001 0.01 0.1 1
Probability[ distance < X ] (in log)
Distance X (in log and normalized by network diameter)
Expansion rates in different networks
T1 (Geographical Distance) - 92824 node-pair samples
T1 (Hop Distance) - 92824 node-pair samples
T2 (Geographical Distance) - 112967 node-pair samples
T2 (Hop Distance) - 112967 node-pair samples
Fig. 3. Comparison of the expansion rates in T1 & T2 (in log-log)
0 5000 10000 15000 20000 25000 30000 35000 40000
0.0
0.2
0.4
0.6
0.8
1.0
CCDF
Distance (km)
Internet latency expansion curve (first 90%)
Internet latency expansion curve (last 10%)
Linear fitting curve
Fig. 4. Internet latency expansion curve of T1 (the first 90%
part) & its fitting curve
IV. LATENCY EXPANSION AT THE CITY LEVEL
Since we assigned geo-locations to all nodes based on
the cities that they belong to, the distance of one node
pair corresponded to that of one city pair. Let N
u
(x)
denote the number of nodes within distance x of u, and
C
u
(x) the number of cities within distance x of u (or the
city where u resides in), and X
i
be the random variable
representing the number of nodes in the city i. Then,
N
u
(x) =
Cu(x)
X
i=1
X
ui
:
Assume all X
i
s have identical and independent distri-
butions. Based on the law of large numbers, N
u
(x) ap-
proaches C
u
(x)E[X] when C
u
(x) is large. Therefore,
N
u
(x)/ C
u
(x):
Therefore, we speculate that the Internet city-level
topology should have the power-law latency expansion
characteristic close to that of the Internet router-level
topology, and the former should be the cause of the
latter. In other words, geography strongly determines
the latency expansion properties of the Internet. By the
Internet city-level topology we mean the city topology in
which cities are considered as points in the geographic
coordinate space, and city connectivity relationship is
based on the neighboring relationship of the routers
inside. That is, there is a link between cities A and B if
and only if there exists at least one router pair (x; y) so
that x is in A, y is in B, and there is a router-level link
between x and y.
To confirm our conjecture, we analyzed the latency
expansions of the Internet city-level topologies in T1 and
T2. Due to their similarity, we only present the results
for T1 here. Figure 5 shows the latency expansion curves
for the Internet on the router level and the Internet on the
city level (602 cities). As before, latency is measured as
the cumulative geographic distance of the path between
the two cities in shortest path routing. As we can see, the
similarity of the two curves is obvious. We conclude that
the Internet city-level topology has power-law expansion
in terms of cumulative geographical distance, and it
decides the Internet router-level latency expansion.
While it is theoretically well-known that a random
subset of nodes in a low-dimensional metric space inher-
its the low expansion rate, it’s also conceivable to embed
some graphs in a low-dimensional metric space while
still preserving their high expansion rate (e.g., embed
a balanced binary tree in a 2-dimensional grid [7]).
Confirmed by our experiments is that the Internet link
topology has low expansion rate after being embedded
into the two-dimensional geographical sphere.
ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 96
1e-05
0.0001
0.001
0.01
0.1
1
10
100
0.0001 0.001 0.01 0.1 1
Probability[ distance < X ] (in log)
Distance X (in log and normalized by network diameter)
Expansion rates in different networks
router level
city level
Internet (Geographical Distance) - 92824 node-pair samples
Internet (Geographical Distance) - 602 cities
Fig. 5. Comparison of latency expansions on router and city level (in
log-log)
V. CONCLUSION
In this note we present our research effort on un-
derstanding the latency expansion characteristic of the
Internet. We find that the Internet has a power-law
latency expansion with an exponent between 1 and 2.
This result has an intriguing connection with, but does
not directly follow from, the fractality of non-uniformly
distributed points in a plane [11]. But this intuition was
not translated into a qualitatively concrete conclusion un-
til we addressed the Internet latency expansion explicitly
with the Internet-scale experiments.
Latency expansion determines the asymptotic perfor-
mance of large-scale distributed systems (web caching,
peer-to-peer systems), whose designs will be influenced
by our finding. Future topology generators that incorpo-
rate link latency will also be impacted by this finding.
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ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 97
ACM SIGCOMM Computer Communications Review Volume 35, Number 1: January 2005 98
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Description
Hui Zhang, Ashish Goel, Ramesh Govindan. "An empirical evaluation of internet latency expansion." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 822 (2004).
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(author),
Govindan, Ramesh
(author),
Zhang, Hui
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USC Computer Science Technical Reports, no. 822 (2004)
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An empirical evaluation of internet latency expansion (
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