Competing contact processes in the Watts-Strogatz network

摘要

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) = {N-1, N-2,..., N-M} of M networks is generated. The network's nodes are decorated with spin-like variables s(i) is an element of {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) = 1 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.