Random graphs as models of networks

TL;DR

This paper reviews advances in random graph models that better mimic real-world networks by incorporating non-Poisson degree distributions and clustering, addressing limitations of classical models like Erdős-Rényi.

Contribution

It introduces generalized random graph models with arbitrary degree distributions and clustering, and discusses their applications to network robustness and epidemic spread.

Findings

Generalized models incorporate realistic degree distributions.

Extensions include clustering to mimic real networks.

Applications demonstrate relevance to robustness and epidemics.

Abstract

The random graph of Erdos and Renyi is one of the oldest and best studied models of a network, and possesses the considerable advantage of being exactly solvable for many of its average properties. However, as a model of real-world networks such as the Internet, social networks or biological networks it leaves a lot to be desired. In particular, it differs from real networks in two crucial ways: it lacks network clustering or transitivity, and it has an unrealistic Poissonian degree distribution. In this paper we review some recent work on generalizations of the random graph aimed at correcting these shortcomings. We describe generalized random graph models of both directed and undirected networks that incorporate arbitrary non-Poisson degree distributions, and extensions of these models that incorporate clustering too. We also describe two recent applications of random graph models to the problems of network robustness and of epidemics spreading on contact networks.