Recovering the Long Range Links in Augmented Graphs Conference Paper uri icon

abstract

  • The augmented graph model, 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 H is 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-range links, which are selected according to the probability distribution ϕ. The augmented graph model enables the analysis of greedy routing in augmented graphs G ∈ (H,ϕ). In greedy routing, each intermediate node handling a message for a target t selects among all its neighbors in G the one that is the closest to t in H and forwards the message to it. This paper addresses the problem of checking whether a given graph G is 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 ∈ (H,ϕ), we aim at extracting the base graph H and the long-range links R out of G. We prove that if H has high clustering coefficient and H has bounded doubling dimension, then a simple local maximum likelihood algorithm enables to partition the edges of G into two sets H′ and R′ such that E(H) ⊆ H′ and the edges in H′ ∖ E(H) are of small stretch, i.e., the map H is not perturbed too greatly by undetected long-range links remaining in H′. The perturbation is actually so small that we can prove that the expected performances of greedy routing in G using the distances in H′ are close to the expected performances of greedy routing using the distances in H. Although this latter result may appear intuitively straightforward, since H′ ⊇ E(H), it is not, as we also show that routing with a map more precise than H may 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 Ω(n 5ε /logn) long-range links of stretch at least Ω(n 1/5 − ε ) for any 0 < ε< 1/5, and thus the map H cannot be recovered with good accuracy.

publication date

  • June 17, 2008