Competing contact processes in the Watts-Strogatz network

  • Marcin Rybak
  • Krzysztof MalarzEmail author
  • Krzysztof Kułakowski
Open Access
Regular Article


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.


Statistical and Nonlinear Physics 


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© The Author(s) 2016

This is an open access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Marcin Rybak
    • 1
  • Krzysztof Malarz
    • 1
    Email author
  • Krzysztof Kułakowski
    • 1
  1. 1.AGH University of Science and Technology, Faculty of Physics and Applied Computer ScienceKrakowPoland

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