Phase transitions in the quadratic contact process on complex networks

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 ($10$) 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.

Figure 1

The solid red (top), dashed, and solid blue (bottom) curves correspond to ${\rho }_{*}={\rho }_{+}$, ${\rho }_{-}$ and 0, respectively, obtained from the mean-field calculation for both QCP types on homogeneous networks.

Figure 2

Steady-state density reached, starting from all vertices occupied, for QCP on homogeneous networks of various sizes $n$.

Figure 3

Steady-state density reached, starting from all vertices occupied, for QCP on power-law networks of various sizes $n$. Note that the $\lambda$ axis is in the log scale.

Figure 4

Steady-state density reached, starting from two different initial densities ${\rho }^{\left(0\right)}$, for QCP on homogeneous networks of size $n={10}^5$. Notice the similarity with the mean-field prediction shown in Fig. 1.

Figure 5

Steady-state density when the birth rate is infinite in the VQCP on various networks of size $n={10}^5$. Note that the ${\rho }^{\left(0\right)}$ axis is in the log scale.

Figure 6

$I(\lambda ,\theta )$ versus $\theta$ near $\lambda ={\lambda }_c$ for various power-law graphs.