Sparse random graphs with clustering

TL;DR

This paper extends the sparse random graph model to include dependent edges by constructing graphs from hypergraphs and replacing hyperedges with complete graphs, allowing for clustering and complex dependence while maintaining mathematical tractability.

Contribution

It introduces a new dependent-edge random graph model based on hypergraph constructions and analyzes its phase transition, degree distribution, and subgraph counts.

Findings

Identifies the critical point for the emergence of a giant component.

Relates the giant component size to a multi-type branching process.

Produces graphs with power-law degree sequences and clustering coefficients.

Abstract

In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.