The simplified self-consistent probabilities method for percolation and its application to interdependent networks

Interdependent networks in areas ranging from infrastructure to economics are ubiquitous in our society, and the study of their cascading behaviors using percolation theory has attracted much attention in recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed, including generating functions and eigenvalues, and others. These different frameworks approach phase transition behaviors from different angles and have been very successful in shaping the different quantities of interest, including critical threshold, size of the giant component, order of phase transition, and the dynamics of cascading. These methods also vary in their mathematical complexity in dealing with interdependent networks that have additional complexity in terms of the correlation among different layers of networks or links. In this work, we review a particular approach of simple, self-consistent probability equations, and we illustrate that this approach can greatly simplify the mathematical analysis for systems ranging from single-layer network to various different interdependent networks. We give an overview of the detailed framework to study the nature of the critical phase transition, the value of the critical threshold, and the size of the giant component for these different systems.