Competing contact processes in the Watts-Strogatz network

Abstract

We investigate two competing contact processes on a set of Watts--Strogatz networks with the clustering coefficient tuned by rewiring. The base for network construction is one-dimensional chain of $N$ sites, where each site $i$ is directly linked to nodes labelled as $i\pm 1$ and $i\pm 2$. So initially, each node has the same degree $k_i=4$. The periodic boundary conditions are assumed as well. For each node $i$ the links to sites $i+1$ and $i+2$ are rewired to two randomly selected nodes so far not-connected to node $i$. An increase of the rewiring probability $q$ influences the nodes degree distribution and the network clusterization coefficient $\mathcal{C}$. For given values of rewiring probability $q$ the set $\mathcal{N}(q)=\{\mathcal{N}_1, \mathcal{N}_2, \cdots, \mathcal{N}_M \}$ of $M$ networks is generated. The network's nodes are decorated with spin-like variables $s_i\in\{S,D\}$. During simulation each $S$ node having a $D$-site in its neighbourhood converts this neighbour from $D$ to $S$ state. Conversely, a node in $D$ state having at least one neighbour also in state $D$-state converts all nearest-neighbours of this pair into $D$-state. The latter is realized with probability $p$. We plot the dependence of the nodes $S$ final density $n_S^T$ on initial nodes $S$ fraction $n_S^0$. Then, we construct the surface of the unstable fixed points in $(\mathcal{C}, p, n_S^0)$ space. The system evolves more often toward $n_S^T=1$ for $(\mathcal{C}, p, n_S^0)$ points situated above this surface while starting simulation with $(\mathcal{C}, p, n_S^0)$ parameters situated below this surface leads system to $n_S^T=0$. The points on this surface correspond to such value of initial fraction $n_S^*$ of $S$ nodes (for fixed values $\mathcal{C}$ and $p$) for which their final density is $n_S^T=\frac{1}{2}$.