Clustering and the hyperbolic geometry of complex networks

TL;DR

This paper analyzes the global clustering coefficient in hyperbolic random graph models of complex networks, providing explicit formulas and showing how it can be tuned by model parameters.

Contribution

It offers a rigorous analysis of clustering in hyperbolic random graphs and derives explicit formulas linking clustering to model parameters.

Findings

Global clustering coefficient characterized mathematically.

Explicit formula for clustering coefficient derived.

Clustering can be tuned via model parameters.

Abstract

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov et al. as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.