The cut metric, random graphs, and branching processes

TL;DR

This paper investigates the component structure of inhomogeneous random graphs and hypergraphs, establishing conditions for the emergence of a giant component and its size using weak convergence assumptions.

Contribution

It generalizes previous phase transition results to broader models by using weak convergence in the cut metric, extending to hypergraphs.

Findings

Criteria for the existence of a giant component

Asymptotic size of the giant component

Generalization to hypergraphs

Abstract

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model we introduced recently, as well as related results of Bollob\'as, Borgs, Chayes and Riordan, all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering.