Susceptibility in inhomogeneous random graphs

TL;DR

This paper investigates the susceptibility in inhomogeneous random graphs, linking it to branching processes, and explores how it can be used to identify phase transition points across various models.

Contribution

It establishes a connection between susceptibility in inhomogeneous random graphs and branching process quantities, providing a framework for analyzing phase transitions.

Findings

Susceptibility relates to branching process parameters.

Provides methods to determine critical points.

Analyzes natural examples of inhomogeneous graphs.

Abstract

We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples.