Recovering the Long-Range Links in Augmented Graphs

  • Authors:
  • Pierre Fraigniaud;Emmanuelle Lebhar;Zvi Lotker

  • Affiliations:
  • CNRS and University Paris Diderot,;CNRS and University Paris Diderot,;Ben Gurion University,

  • Venue:
  • SIROCCO '08 Proceedings of the 15th international colloquium on Structural Information and Communication Complexity
  • Year:
  • 2008

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Abstract

The augmented graphmodel, as introduced by Kleinberg (STOC 2000), is an appealing model for analyzing navigability in social networks. Informally, this model is defined by a pair (H,φ), where His a graph in which inter-node distances are supposed to be easy to compute or at least easy to estimate. This graph is "augmented" by links, called long-rangelinks, which are selected according to the probability distribution φ. The augmented graph model enables the analysis of greedy routingin augmented graphs G茂戮驴 (H,φ). In greedy routing, each intermediate node handling a message for a target tselects among all its neighbors in Gthe one that is the closest to tin Hand forwards the message to it.This paper addresses the problem of checking whether a given graph Gis an augmented graph. It answers part of the questions raised by Kleinberg in his Problem 9 (Int. Congress of Math. 2006). More precisely, given G茂戮驴 (H,φ), we aim at extracting the base graph Hand the long-range links Rout of G. We prove that if Hhas high clustering coefficient and Hhas bounded doubling dimension, then a simple local maximum likelihood algorithm enables to partition the edges of Ginto two sets H茂戮驴 and R茂戮驴 such that E(H) 茂戮驴 H茂戮驴 and the edges in H茂戮驴 茂戮驴 E(H) are of small stretch, i.e., the map His not perturbed too greatly by undetected long-range links remaining in H茂戮驴. The perturbation is actually so small that we can prove that the expected performances of greedy routing in Gusing the distances in H茂戮驴 are close to the expected performances of greedy routing using the distances in H. Although this latter result may appear intuitively straightforward, since H茂戮驴 茂戮驴 E(H), it is not, as we also show that routing with a map more precise than Hmay actually damage greedy routing significantly. Finally, we show that in absence of a hypothesis regarding the high clustering coefficient, any local maximum likelihood algorithm extracting the long-range links can miss the detection of at least 茂戮驴(n5茂戮驴/logn) long-range links of stretch at least 茂戮驴(n1/5 茂戮驴 茂戮驴) for any 0 茂戮驴Hcannot be recovered with good accuracy.