Robustness of a Tree-like Network of Interdependent Networks
Jianxi Gao, S. V. Buldyrev, S. Havlin, and H. E. Stanley

TL;DR
This paper develops an analytical framework to study the robustness and cascading failures in interdependent networks, revealing that regular networks are more resilient than Erdős-Rényi networks when coupled.
Contribution
It introduces a general exact percolation law for a network of interdependent networks and compares the robustness of coupled ER and RR networks.
Findings
RR networks exhibit higher robustness than ER networks.
First order percolation transition occurs for multiple coupled networks.
No minimum degree threshold for RR networks' stability.
Abstract
In reality, many real-world networks interact with and depend on other networks. We develop an analytical framework for studying interacting networks and present an exact percolation law for a network of interdependent networks (NON). We present a general framework to study the dynamics of the cascading failures process at each step caused by an initial failure occurring in the NON system. We study and compare both coupled Erd\H{o}s-R\'{e}nyi (ER) graphs and coupled random regular (RR) graphs. We found recently [Gao et. al. arXive:1010.5829] that for an NON composed of ER networks each of average degree , the giant component, , is given by where is the initial fraction of removed nodes. Our general result coincides for with the known Erd\H{o}s-R\'{e}nyi second-order phase transition at a threshold,…
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