Abstract
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.
Highlights
Many multiplex networks are embedded in space, with links more likely to exist between nearby nodes than distant nodes
In models of power grid topology, links are formed with the m nearest neighbors, statically [33] or as a generative model [34]
In addition to providing a way to measure the effects of spatiality on multiplex networks, our model presents an alternative and more realistic way to study interdependent networks
Summary
Many multiplex networks are embedded in space, with links more likely to exist between nearby nodes than distant nodes. Previous research on the robustness of spatially embedded interdependent networks considered coupled lattices with dependency links of geographic length up to r, a system parameter [51]. We focus on the case in which the multiplex consists of two layers, each with the same number of nodes, characteristic link length ζ and average degree k
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