Frozen percolation on inhomogeneous random graphs

Abstract

Mean-field frozen percolation is a random graph-valued process, which adjusts the dynamics of the classical Erdos-Renyi process with an additional mechanism to 'freeze' potential giant components before they can form. It is known to exhibit self-organised criticality from a wide class of initial graphs. We show that a family of inhomogeneous random graphs with finitely-many types form a stable class under these dynamics. We study how the survival of a vertex depends on its initial type, and establish a hydrodynamic limit for the process recording surviving vertices of each type, in terms of multitype branching processes which approximate the graphs. The parameters of these branching processes are eventually critical, and their evolution in time is described by solutions to an unusual class of differential equations driven by Perron-Frobenius eigenvectors.