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

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.

💡 Research Summary

The paper introduces a novel framework for studying cascading failures in two interdependent networks when each node may receive support from multiple counterpart nodes. Unlike earlier models that assume a one‑to‑one dependency, the authors consider “multiple support‑dependent relations”: a node in network A (or B) remains functional as long as it has at least one support link from network B (or A). Both networks are initially modeled as Erdős–Rényi (ER) random graphs with sizes N^A and N^B, average degrees a and b, and with c₀^{AB} N^B directed support links from A to B and c₀^{BA} N^A links from B to A.

The failure process consists of two stages. First, a random attack removes fractions (1‑R^A) of nodes from A and (1‑R^B) from B. Second, a cascade of failures unfolds: any node that loses all its support links or becomes disconnected from the giant component of its own network is removed, which may in turn strip support from nodes in the other network, and so on. The authors formalize this iterative process as a set of coupled recursive equations for the surviving fractions μ_n^A and μ_n^B at iteration n, and they analyze the fixed point (n → ∞) to obtain the final giant‑component sizes μ_∞^A and μ_∞^B.

The main analytical result is a pair of coupled transcendental equations:

 μ_∞^A = R^A