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

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} ∈ { *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*. Then, we construct the surface of the unstable fixed points in (𝓒,

_{S}^{0}*p, n*) space. The system evolves more often toward

_{S}^{0}*n*for (𝓒,

_{S}^{T}*p, n*) points situated above this surface while starting simulation with (𝓒,

_{S}^{0}*p, n*) parameters situated below this surface leads system to

_{S}^{0}*n*. The points on this surface correspond to such value of initial fraction

_{S}^{T}=0*n*of

_{S}^{*}*S*nodes (for fixed values 𝓒 and

*p*) for which their final density is

*n*=1/2.

_{S}^{T}Key words: Statistical and Nonlinear Physics

*© The Author(s) 2016. This article is published with open access at
Springerlink.com*

This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted
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properly cited.