General and exact approach to percolation on random graphs

TL;DR

This paper develops a comprehensive theoretical framework for analyzing percolation on complex random graphs, providing exact solutions for component distributions and phase transitions, including in interdependent and clustered networks.

Contribution

It introduces a set of iterative equations for exact component distributions, defines a general graph ensemble for exact percolation analysis, and extends the approach to interdependent graphs with phase transition insights.

Findings

Exact component size distributions for finite and infinite graphs.

Percolation thresholds and component compositions derived analytically.

Clustering amplifies discontinuous phase transition effects.

Abstract

We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day, and also that permits to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite size limit using probability generating functions [i.e., the percolation threshold, the size and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs---whose most striking feature is the emergence of an extensive component via a discontinuous phase transition---in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.