Duality in inhomogeneous random graphs, and the cut metric

TL;DR

This paper establishes a duality principle for a broad class of inhomogeneous random graphs, linking supercritical and subcritical regimes via the cut metric, simplifying the analysis of their component structures.

Contribution

It proves a duality principle for inhomogeneous random graphs with edge probabilities converging in the cut metric, generalizing previous models.

Findings

Duality principle holds for inhomogeneous random graphs

Applicable to models with edge probabilities converging in cut metric

Simplifies analysis of component structures in complex networks

Abstract

The classical random graph model $G(n,\lambda/n)$ satisfies a `duality principle', in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric.