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Interdependent networks are ubiquitous in our society, ranging frominfrastructure to economics, and the study of their cascading behaviors usingpercolation theory has attracted much attention in the recent years. To analyzethe percolation phenomena of these systems, different mathematical frameworkshave been proposed including generating functions, eigenvalues among someothers. These different frameworks approach the phase transition behaviors fromdifferent angles, and have been very successful in shaping the differentquantities of interest including critical threshold, size of the giantcomponent, order of phase transition and the dynamics of cascading. Thesemethods also vary in their mathematical complexity in dealing withinterdependent networks that have additional complexity in terms of thecorrelation among different layers of networks or links. In this work, wereview a particular approach of simple self-consistent probability equations,and illustrate that it can greatly simplify the mathematical analysis forsystems ranging from single layer network to various different interdependentnetworks. We give an overview on the detailed framework to study the nature ofthe critical phase transition, value of the critical threshold and size of thegiant component for these different systems.

A Simplified Self-Consistent Probabilities Framework to Characterize Percolation Phenomena on Interdependent Networks : An Overview