Percolation on interacting networks

TL;DR

This paper develops a mathematical framework using generating functions to analyze percolation in systems of multiple interacting networks, revealing how inter-network connections influence large-scale connectivity.

Contribution

It introduces exact formulas for percolation thresholds in interacting networks with arbitrary degree distributions, extending percolation theory to complex multi-network systems.

Findings

Percolation threshold can be significantly lowered by inter-network connections.

Explicit evaluation for two networks shows the impact of degree distributions.

Applications demonstrate relevance to communication and socio-technical systems.

Abstract

Most networks of interest do not live in isolation. Instead they form components of larger systems in which multiple networks with distinct topologies coexist and where elements distributed amongst different networks may interact directly. Here we develop a mathematical framework based on generating functions for analyzing a system of L interacting networks given the connectivity within and between networks. We derive exact expressions for the percolation threshold describing the onset of large-scale connectivity in the system of networks and each network individually. These general expressions apply to networks with arbitrary degree distributions and we explicitly evaluate them for L=2 interacting networks with a few choices of degree distributions. We show that the percolation threshold in an individual network can be significantly lowered once "hidden" connections to other networks are considered. We show applications of the framework to two real-world systems involving communications networks and socio-tecnical congruence in software systems.