Competing contact processes in the Watts-Strogatz network

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 ± 1 and i ± 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 𝓒. For given values of rewiring probability q the set 𝓝(q)={𝓝1,𝓝2,...,𝓝 M } of M networks is generated. The network’s nodes are decorated with spin-like variables s i ∈ { 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 (𝓒, p, n S 0) space. The system evolves more often toward n S T for (𝓒, p, n S 0) points situated above this surface while starting simulation with (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 𝓒 and p) for which their final density is n S T=1/2.