Percolation on networks with weak and heterogeneous dependency

TL;DR

This paper introduces a percolation model on networks with weak and heterogeneous node dependencies, revealing that dependency heterogeneity enhances robustness and influences the nature of phase transitions.

Contribution

It develops a novel percolation model accounting for variable dependency strengths and analyzes how their distribution affects network robustness and transition types.

Findings

Heterogeneous dependency strengths increase network robustness.

Both continuous and discontinuous percolation transitions can occur.

Transition points depend on moments of dependency strength distribution.

Abstract

In real networks, the dependency between nodes is ubiquitous; however, the dependency is not always complete and homogeneous. In this paper, we propose a percolation model with weak and heterogeneous dependency; i.e., dependency strengths could be different between different nodes. We find that the heterogeneous dependency strength will make the system more robust, and for various distributions of dependency strengths both continuous and discontinuous percolation transitions can be found. For Erd\H{o}s-R\'{e}nyi networks, we prove that the crossing point of the continuous and discontinuous percolation transitions is dependent on the first five moments of the dependency strength distribution. This indicates that the discontinuous percolation transition on networks with dependency is determined not only by the dependency strength but also by its distribution. Furthermore, in the area of the continuous percolation transition, we also find that the critical point depends on the first and second moments of the dependency strength distribution. To validate the theoretical analysis, cases with two different dependency strengths and Gaussian distribution of dependency strengths are presented as examples.