Percolation on interdependent networks with a fraction of antagonistic interactions

TL;DR

This paper analyzes how the coexistence of interdependent and antagonistic interactions affects percolation transitions in two networks, revealing bistability, phase transition types, and robustness thresholds.

Contribution

It characterizes the phase diagram of percolation in interdependent networks with antagonistic nodes, highlighting bistability and the impact of antagonistic interaction fraction.

Findings

Bistability of steady states in certain parameter regions.

Presence of both first and second order phase transitions.

A critical antagonistic fraction q_c=2/3 reduces percolation regions.

Abstract

Recently, the percolation transition has been characterized on interacting networks both in presence of interdependent and antagonistic interactions. Here we characterize the phase diagram of the percolation transition in two Poisson interdependent networks with a percentage q of antagonistic nodes. We show that this system can present a bistability of the steady state solutions, and both first, and second order phase transitions. In particular, we observe a bistability of the solutions in some regions of the phase space also for a small fraction of antagonistic interactions 0<q<0.4. Moreover, we show that a fraction q>q_c=2/3 of antagonistic interactions is necessary to strongly reduce the region in phase-space in which both networks are percolating. This last result suggests that interdependent networks are robust to the presence of antagonistic interactions. Our approach can be extended to multiple networks, and to complex boolean rules for regulating the percolation phase transition.