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Robustness of a Network of Networks
Jianxi Gao, Sergey V. Buldyrev, Shlomo Havlin, H. Eugene Stanley
TL;DR
The paper addresses robustness in interacting networks, extending analysis beyond isolated networks. It develops an analytical framework for interdependent networks and derives an exact percolation law showing cascading failures and first-order transitions when n > 1.
Problem
Most network research studies isolated networks, despite real-world infrastructures depending on one another.
Method
The paper develops a mathematical framework for a network of n coupled randomly connected networks with bidirectional one-to-one dependency links.
Results
For n interdependent ER networks, the exact percolation law generalizes single-network ER percolation; n > 1 produces cascading failures and a first-order transition.
Takeaways & Limitations
Classical single-network percolation is the n = 1 limiting case of a broader interdependent-network percolation theory, while vulnerability increases with n.
Takeaways & Limitations
The analytical results for equal-degree ER networks assume all n networks have the same average degree, and random dependency loops can make the system unstable.
Abstract
from arXiv · showhide
Almost all network research has been focused on the properties of a single network that does not interact and depends on other networks. In reality, many real-world networks interact with other networks. Here we develop an analytical framework for studying interacting networks and present an exact percolation law for a network of $n$ interdependent networks. In particular, we find that for $n$ Erdős-Rényi networks each of average degree $k$, the giant component, $P_{\infty}$, is given by $P_{\infty}=p[1-\exp(-kP_{\infty})]^n$ where $1-p$ is the initial fraction of removed nodes. Our general result coincides for $n=1$ with the known Erdős-Rényi second-order phase transition for a single network. For any $n \geq 2$ cascading failures occur and the transition becomes a first-order percolation transition. The new law for $P_{\infty}$ shows that percolation theory that is extensively studied in physics and mathematics is a limiting case ($n=1$) of a more general general and different percolation law for interdependent networks.