@MISC{Chen_wenjiefang, author = {Wei Chen and Ecole Normale and Guangda Hu and Michael W. Mahoney}, title = {Wenjie Fang}, year = {} }

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Abstract

Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δ-hyperbolicity, and we establish several positive and negative results for small-world and tree-like random graph models. First, we study the hyperbolicity of the class of Kleinberg small-world random graphs KSW (n, d, γ), where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the small-world parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB(u, v)γ with dB(u, v) being the grid distance from u to v in the base grid B. We show that when γ = d, the parameter value allowing efficient decentralized routing in Kleinberg’s small-world network, with probability 1 − o(1) the hyperbolic δ is Ω((log n) 11.5(d+1)+ε) for any ε> 0 independent of n. Comparing to the diameter of Θ(log n) in this case, it indicates that hyperbolicity is not significantly improved comparing to graph diameter even when the long-range connections greatly improves decentralized navigation. We also show that for other values of γ the hyperbolic δ is either at the same level or very close to the graph diameter, indicating poor hyperbolicity in these graphs as well.