Cascading failures in coupled networks with both inner-dependency and inter-dependency links

TL;DR

This paper investigates how the mix of inner- and inter-dependency links affects the robustness and phase transition type in coupled networks, revealing conditions for continuous or discontinuous percolation transitions.

Contribution

It introduces a model analyzing the effects of dependency link types on network robustness and identifies a critical average degree influencing transition behavior.

Findings

High dependency density can cause discontinuous transitions.

Balancing inner- and inter-dependency links enhances robustness.

A critical average degree determines the transition type.

Abstract

We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.