Phase transitions in the quadratic contact process on complex networks

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 λ and infected individuals recover (1→0) at rate 1. In the QCP, a combination of two 1's is required to effect a 0→1 change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. We define two versions of the QCP: vertex-centered (VQCP) and edge-centered (EQCP) with birth events 1-0-1→1-1-1 and 1-1-0→1-1-1, respectively, where "-" represents an edge. We investigate the effects of network topology by considering the QCP on random regular, Erdős-Rényi, 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 Erdős-Rényi 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.