Cascade of failures in coupled network systems with multiple support-dependent relations

TL;DR

This paper models and analyzes the cascading failures in two coupled networks with support-dependent relations, deriving percolation laws for the surviving giant components after random attacks, and validates the theory with simulations on Erdős-Rényi and scale-free networks.

Contribution

The paper introduces a new analytical framework for cascading failures in coupled networks with multiple support-dependent links, extending previous models to include random support relations.

Findings

Derived explicit percolation laws for coupled networks' giant components.

Showed that increasing support links reduces cascading failures.

Validated theoretical predictions with simulations on ER and scale-free networks.

Abstract

We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependent relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support node in the other network. If both networks A and B are Erd\H{o}s-R\'enyi networks, A and B, with (i) sizes $N^A$ and $N^B$, (ii) average degrees $a$ and $b$, and (iii) $c^{AB}_0N^B$ support links from network A to B and $c^{BA}_0N^B$ support links from network B to A, we find that under random attack with removal of fractions $(1-R^A)N^A$ and $(1-R^B)N^B$ nodes respectively, the percolating giant components of both networks at the end of the cascading failures, $\mu^A_\infty$ and $\mu^B_\infty$, are given by the percolation laws $\mu^A_\infty = R^A [1-\exp{({-c^{BA}_0\mu^B_\infty})}] [1-\exp{({-a\mu^A_\infty})}]$ and $\mu^B_\infty = R^B [1-\exp{({-c^{AB}_0\mu^A_\infty})}] [1-\exp{({-b\mu^B_\infty})}]$. In the limit of $c^{BA}_0 \to \infty$ and $c^{AB}_0 \to \infty$, both networks become independent, and the giant components are equivalent to a random attack on a single Erd\H{o}s-R\'enyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.