Random Hyperbolic Graphs: Degree Sequence and Clustering

Abstract

In the last decades, the study of models for large real-world networks has been a very popular and active area of research. A reasonable model should not only replicate all the structural properties that are observed in real world networks (for example, heavy tailed degree distributions, high clustering and small diameter), but it should also be amenable to mathematical analysis. There are plenty of models that succeed in the first task but are hard to analyze rigorously. On the other hand, a multitude of proposed models, like classical random graphs, can be studied mathematically, but fail in creating certain aspects that are observed in real-world networks.Recently, Papadopoulos, Krioukov, Boguñá and Vahdat [INFOCOM'10] introduced a random geometric graph model that is based on hyperbolic geometry. The authors argued empirically and by some preliminary mathematical analysis that the resulting graphs have many of the desired properties. Moreover, by computing explicitly a maximum likelihood fit of the Internet graph, they demonstrated impressively that this model is adequate for reproducing the structure of real graphs with high accuracy.In this work we initiate the rigorous study of random hyperbolic graphs. We compute exact asymptotic expressions for the expected number of vertices of degree k for all k up to the maximum degree and provide small probabilities for large deviations. We also prove a constant lower bound for the clustering coefficient. In particular, our findings confirm rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.

A Bound for the Diameter of Random Hyperbolic Graphs

“…The main problem we address in this work is the natural question, explicitly stated in [GPP12,page 6], that asks to determine the expected diameter of the giant component of a random hyperbolic graph G chosen according to G α,C (n) for 1 2 < α < 1. We look at this range, since for α < 1 2 a.a.s.…”

“…In words, the random hyperbolic graph model is a simple variant of the uniform distribution of n vertices within a disc of radius R of the hyperbolic plane, where two vertices are connected if their hyperbolic distance is at most R. Formally, the random hyperbolic graph model G α,C (n) is defined in [GPP12] as described next: for α > 1 2 , C ∈ R, n ∈ N, set R = 2 ln n + C, and build G = (V, E) with vertex set V = [n] as follows:…”

On the Diameter of Hyperbolic Random Graphs

“…Although this paper is self-contained, we recommend to a reader who is unfamiliar with the notion of hyperbolic random graphs the more thorough investigations [16,19].…”

“…∆ϕ u,v := cos −1 (cos(ϕ u − ϕ v )) π. This results in a graph whose degree distribution follows a power law with exponent β = 2α + 1, if α 1 2 , and β = 2 otherwise [16]. Since most real-world networks have been shown to have a power law exponent 2 < β < 3, we assume throughout the paper that 1 2 < α < 1.…”

“…Since most real-world networks have been shown to have a power law exponent 2 < β < 3, we assume throughout the paper that 1 2 < α < 1. Gugelmann et al [16] proved that the average degree in this model is then δ = (1 + o(1))…”