Percolation of networks with directed dependency links

TL;DR

This paper studies how directed dependency links affect the percolation phase transitions in networks, revealing that key transition points depend on the proportion of nodes with minimal dependencies, using a probabilistic approach.

Contribution

It introduces a probabilistic method to analyze percolation in networks with directed dependency links, highlighting how minimal dependency proportions influence criticality and phase transition types.

Findings

Critical point depends on nodes with no dependencies.

Triple point determined by nodes depending on at most one node.

Results illustrated with Erdős-Rényi networks.

Abstract

The self-consistent probabilistic approach has proven itself powerful in studying the percolation behavior of interdependent or multiplex networks without tracking the percolation process through each cascading step. In order to understand how directed dependency links impact criticality, we employ this approach to study the percolation properties of networks with both undirected connectivity links and directed dependency links. We find that when a random network with a given degree distribution undergoes a second-order phase transition, the critical point and the unstable regime surrounding the second-order phase transition regime are determined by the proportion of nodes that do not depend on any other nodes. Moreover, we also find that the triple point and the boundary between first- and second-order transitions are determined by the proportion of nodes that depend on no more than one node. This implies that it is maybe general for multiplex network systems, some important properties of phase transitions can be determined only by a few parameters. We illustrate our findings using Erdos-Renyi (ER) networks.