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On the Curvature of the Internet and its usage
for
Overlay Construction and Distance Estimation
Yuval Shavitt and Tomer Tankel
Abstract
— It was noted in recent years that the Internet
structure resembles a star with a highly connected core
and long stretched tendrils. In this work we present a new
quantity, the Internet geometric curvature, that captures
the above observation by a single number. We embed the
Internet distance metric in a hyperbolic space with an
optimal curvature and achieve an accuracy better than
achieved before for the Euclidean space. This proves our
hypothesis regarding the internet curvature. We demon
strate the strength of our embedding with two applications:
selecting the closest server and building an application level
multicast tree.
I. INTRODUCTION
The internet structure has been the subject of many
recent works. Researchers have looked at various features
of the Internet graph, and proposed theoretical models to
describe its evolvement. Faloutsos
et al.
[1] experimen
tally discovered that the degree distribution of the Internet
AS and router level graphs obey a power law. Barab´asi
and Albert [2], [3] developed an evolutional model of
preferential attachment, that can be used for generating
topologies with powerlaw degree distributions. The Inter
net AS structure was shown to have a core in the middle
and many tendrils connected to it [4], [5]. A more detailed
descriptions is that around the core there are several rings
of nodes all have tendrils of varying length attached to
them. The average node degree decreases as one moves
away from the core.
In this paper we identi±ed a new characteristic of the
Internet graph, its
curvature
. We use this curvature to
better represent the Internet distance map in a geometric
space. Using this realistic representation we were able to
improve performance of three applications: Delay estima
tion (which can be used for QoS threshold estimation),
Server Selection, and Application Level Multicast.
The geometry of a distance matrix can be represented
by mapping its nodes in a real geometric space. Such a
mapping, called
embedding
, is designed to preserve the
distance between any pair of network nodes close to the
distance between their geometric images. The
symmetric
pair distortion
is de±ned for each pair as the maximum
of the ratio between the original and geometric distance
a
b
a
bb
b
a
a
CD=2a+b
AC=2a+2b
a
a
b
b
b
b
a
a
D
AB
C
CD=2a+b
AC=2a+2b
A
C
D
B
Fig. 1.
ab Graph in
D
2
and its inverse. An input metric can be embedded in two
classes of algorithms
1) All pair (AP) embedding. The entire
n
nodes met
ric, that is comprising
n
(
n
−
1)
/
2
distance pairs, is
embedded at once.
2) Two phase (TP) embedding. First, a small subset of
t
≤
15
nodes, called
Tracers
, is embedded, consid
ering all
t
(
t
−
1)
/
2
pair distances. The coordinates
of the rest of the nodes are calculated from their
0780383567/04/$20.00 (C) 2004 IEEE
IEEE INFOCOM 2004
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View Full Documentdistance to several
nearest
Tracers by minimizing
the symmetric distortion of these nodeTracer pairs.
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 Spring '08
 Gupta

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