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^′ {set minus} 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 (n^{5 ε} / log n) long-range links of stretch Ω (n^{1 / 5 - ε}) for any 0 < ε < 1 / 5, and thus the map H cannot be recovered with good accuracy.