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

TL;DR

This paper reviews a simplified self-consistent probabilities framework for analyzing percolation phenomena in interdependent networks, making the mathematical study of phase transitions and cascading failures more accessible.

Contribution

It introduces and illustrates a simplified approach using self-consistent probability equations to analyze complex interdependent networks, reducing mathematical complexity.

Findings

The framework effectively characterizes critical thresholds and phase transition nature.

It simplifies analysis across various network types.

The approach aids in understanding cascading behaviors.

Abstract

Interdependent networks are ubiquitous in our society, ranging from infrastructure to economics, and the study of their cascading behaviors using percolation theory has attracted much attention in the recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed including generating functions, eigenvalues among some others. These different frameworks approach the 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 illustrate that it can greatly simplify the mathematical analysis for systems ranging from single layer network to various different interdependent networks. We give an overview on the detailed framework to study the nature of the critical phase transition, value of the critical threshold and size of the giant component for these different systems.