Eur. Phys. J. B (2016) 89: 145
https://doi.org/10.1140/epjb/e2016-70135-2

Regular Article

Competing contact processes in the Watts-Strogatz network

AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Krakow, Poland

^a e-mail: malarz@agh.edu.pl

Received: 4 March 2016
Received in final form: 15 April 2016
Published online: 8 June 2016

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 ± 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_S⁰. Then, we construct the surface of the unstable fixed points in (𝓒, p, n_S⁰) space. The system evolves more often toward n_S^T for (𝓒, p, n_S⁰) points situated above this surface while starting simulation with (𝓒, p, n_S⁰) 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.