Phase Transitions in the Quadratic Contact Process on Complex Networks

TL;DR

This paper extends the quadratic contact process to complex networks, analyzing phase transitions and bistability on different graph types through simulations and mean field calculations.

Contribution

It introduces vertex and edge centered versions of the QCP on complex networks and characterizes their phase transition behaviors.

Findings

Discontinuous phase transition with bistability on regular and Erdős-Rényi graphs.

Continuous transition on power law networks.

Critical birth rate positive for regular and Erdős-Rényi, zero for power law.

Abstract

The quadratic contact process (QCP) is a natural extension of the well studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate $\lambda$ and infected individuals recover ($1 \longrightarrow 0$) at rate 1. In the QCP, a combination of two 1's is required to effect a $0 \longrightarrow 1$ change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. \comment{as a model for the change in a population through sexual reproduction and death.} We define two versions of the QCP -- vertex centered (VQCP) and edge centered (EQCP) with birth events $1-0-1 \longrightarrow 1-1-1$ and $1-1-0 \longrightarrow 1-1-1$ respectively, where `$-$' represents an edge. We investigate the effects of network topology by considering the QCP on random regular, Erd\H{o}s-R\'{e}nyi and power law random graphs. We perform mean field calculations as well as simulations to find the steady state fraction of occupied vertices as a function of the birth rate. We find that on the random regular and Erd\H{o}s-R\'{e}nyi graphs, there is a discontinuous phase transition with a region of bistability, whereas on the heavy tailed power law graph, the transition is continuous. The critical birth rate is found to be positive in the former but zero in the latter.