09_1 - On the Curvature of the Internet and its usage for...

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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 power-law 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. a-b 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 0-7803-8356-7/04/$20.00 (C) 2004 IEEE IEEE INFOCOM 2004
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distance to several nearest Tracers by minimizing the symmetric distortion of these node-Tracer pairs.
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09_1 - On the Curvature of the Internet and its usage for...

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