Competing contact processes in the Watts-Strogatz network

Abstract

We investigate two competing contact processes on a set of Watts–Strogatz networks withthe clustering coefficient tuned by rewiring. The base for network construction isone-dimensional chain of N sites, where each site i is directly linked tonodes labelled as i ±1 and i ±2. So initially, each node has the same degree ki =4. The periodic boundary conditions are assumed as well. For each nodei the linksto sites i +1 and i +2 are rewired to two randomly selected nodes so far not-connected tonode i. Anincrease of the rewiring probability q influences the nodes degree distribution and thenetwork clusterization coefficient C. For given values of rewiring probabilityq the set N(q)={N1,N2,..,NM} of M networks is generated. The network’s nodes aredecorated with spin-like variables si ∈ { S,D}. During simulation each S node having a D-site in its neighbourhoodconverts this neighbour from D to S state. Conversely, a node in D state having at least oneneighbour also in state D-state converts all nearest-neighbours of this pairinto D-state. The latter is realized with probabilityp. We plotthe dependence of the nodes S final density nST on initial nodes S fraction nS0. Then, we construct the surface of the unstable fixedpoints in (C, p, nS0) space. The system evolves more often toward nST for (C, p, nS0) points situated above this surface while startingsimulation with (C, p, nS0) parameters situated below this surface leads system to nST=0. The points on this surface correspond to such value ofinitial fraction nS* of S nodes (for fixed values C and p) for which their final density is nST=1/2.