Recovering the long-range links in augmented graphs

Abstract

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 ( 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.