Recovering the Long-Range Links in Augmented Graphs

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.