Enhanced robustness of single-layer networks with redundant dependencies

TL;DR

This paper investigates how relaxing dependency rules in single-layer networks enhances their robustness, revealing complex phase transitions and critical phenomena in Erd"os-Rényi and scale-free networks.

Contribution

It introduces a weak dependency model allowing nodes to function if at least one dependency neighbor is active, and analyzes its impact on network robustness and phase transitions.

Findings

Identification of a 'cusp' point where the system remains stable regardless of dependency density.

Discovery of hybrid percolation transitions, both continuous and discontinuous, in different network types.

Observation that correlated degrees in scale-free networks enhance resilience, especially for hubs.

Abstract

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: nodes with dependency neighbours may continue to function if at least one of their dependency neighbours is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erd\"os-R\'enyi and random scale-free networks with an underlying Erd\"os-R\'enyi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erd\"os-R\'enyi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity $S-S_c \propto (p-p_c)^{1/4}$. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.