Recovering the long-range links in augmented graphs

The augmented graph model, as introduced in Kleinberg, STOC (2000) [23], 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, that 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 a high clustering coefficient and H has bounded doubling dimension, then a simple local maximum likelihood algorithm enables us 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 the absence of a hypothesis regarding the high clustering coefficient, any local maximum likelihood algorithm extracting the long-range links can miss the detection of Ω(n5ε/logn) long-range links of stretch Ω(n1/5−ε) for any 0<ε<1/5, and thus the map H cannot be recovered with good accuracy.