Percolation on interacting networks with feedback-dependency links

TL;DR

This paper develops a mathematical framework to analyze percolation in coupled networks with feedback-dependency links, revealing how network robustness varies with coupling strength, average degrees, and feedback dependencies.

Contribution

It introduces a novel analytical and numerical approach to study percolation in networks with feedback dependencies, highlighting their impact on system robustness and phase transition types.

Findings

Low inter-network connectivity leads to different phase transition behaviors.

Increasing intra-network average degree enhances robustness under strong coupling.

Feedback dependency links significantly increase system vulnerability.

Abstract

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here we develop a mathematical framework and study numerically and analytically percolation of interacting networks with feedback-dependency links. We find that when nodes of between networks are lowly connected, the system undergoes from second order transition through hybrid order transition to first order transition as coupling strength increases. And, as average degree of each inter-network increases, first order region becomes smaller and second-order region becomes larger but hybrid order region almost keep constant. Especially, the results implies that average degree \bar{k} between intra-networks has a little influence on robustness of system for weak coupling strength, but for strong coupling strength corresponding to first order transition system become robust as \bar{k} increases. However, when average degree k of inter-network is increased, the system become robust for all coupling strength. Additionally, when nodes of between networks are highly connected, the hybrid order region disappears and the system first order region becomes larger and secondorder region becomes smaller. Moreover, we find that the existence of feedback dependency links between interconnecting networks makes the system extremely vulnerable by comparing non-feedback condition for the same parameters.