Lack of Hyperbolicity in Asymptotic Erdös–Renyi Sparse Random Graphs

Abstract

In this work we prove that the giant component of the Erdös-Renyi random graph G(n, c/n) for c a constant greater than 1 (sparse regime), is not Gromov δ-hyperbolic for any δ with probability tending to one as n → ∞. As a corollary we provide an alternative proof that the giant component of G(n, c/n) when c > 1 has zero spectral gap almost surely as n → ∞.

On Computing the Hyperbolicity of Real-World Graphs

“…In general, it is still not clear whether the hyperbolicity value is small in all real-world networks (as it seems from [19,2]), or it is a characteristic of specific networks (as it seems from [1]). Finally, the hyperbolicity of random graphs has been analyzed in the case of several random graph models, such as the Erdös-Renyi model [24] and the Kleinberg model [6]. Moreover, in this latter paper, it is stated that the design of more efficient exact algorithms for the computation of the hyperbolicity would be of interest.…”

“…The simplest model considered is the Erdös-Renyi random graph G n,m , that is, we choose a graph with n nodes and m edges uniformly at random. In this model, it has been proved that the hyperbolicity tends to infinity [24], and, if m is "much bigger than n", exact asymptotics for δ have been computed [22]. Instead, the hyperbolicity of sparse Erdös-Renyi graphs is not known, and it is mentioned as an open problem in [22].…”

“…In this context, a significant role is played by the hyperbolic structure underlying a complex network, that is usually not present in random graphs [24,6]. For instance, if we draw points from a hyperbolic space and we connect nearby points, This work has been supported in part by the Italian Ministry of Education, University, and Research under PRIN 2012C4E3KT national research project AMANDA (Algorithmics for MAssive and Networked Data), and by ANR project Stint under reference ANR-13-BS02-0007, ANR program "Investments for the Future" under reference ANR-11-LABX-0031-01.…”

Minimum node degree and k-connectivity in wireless networks with unreliable links

“…The unreliable node-to-node links with probability p n of being active and probability (1 − p n ) of being inactive correspond to a random graph model called Erdős-Rényi graph [19] [20] [21] G(n, p n ). Early papers [19], [22] introduce G(n, p n ) defined on n nodes such that an edge between any two nodes exists independently with probability p n , where p n is a function of n. Erdős and Rényi [19] derive that when p n is ln n+αn n , graph G(n, p n ) is almost surely connected (resp., not connected) if lim n→∞ α n = ∞ (resp., lim n→∞ α n = −∞).…”

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

“…We refer the reader to [21,24,29,47], and references therein, for details on the motivating network applications. There has been a recent surge of interest in studying the hyperbolicity of various classes of random networks including small world networks [42,47], Erdős-Rényi random graphs [30], and random graphs with expected degrees [43].…”