On the Hyperbolicity of Small-World and Tree-Like Random Graphs

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) 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) 1 1.5(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. Next we study a class of tree-like graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.