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 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.
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References
T.M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer-Verlag, Berlin, 1999)
E.T. Harris, Ann. Probab. 2, 969 (1974)
J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge UP, Cambridge, 1999)
M.M. de Oliveira, R.V. Dos Santos, R. Dickman, Phys. Rev. E 86, 011121 (2012)
B. Oborny, G. Meszéna, G. Szabó, Oikos 109, 291 (2005)
F. Giargulo, S. Huet, Europhys. Lett. 91, 58004, (2010)
T.M. Liggett, Ann. Probab. 24, 1675, (1996)
M. Henkel, H. Hinrichsen, J. Phys. A 37, R117 (2004)
G. Ódor, R. Pastor-Satorras, Phys. Rev. E 86, 026117 (2012)
I. Jensen, Phys. Rev. Lett. 70, 1465 (1993)
M. Rybak, K. Kułakowski, Int. J. Mod. Phys. C 24, 1350012 (2013)
M. Rybak, K. Kułakowski, Int. J. Mod. Phys. C 27, 1692002 (2016)
M. Rybak, K. Kułakowski, Acta Phys. Pol. A 44, 1007 (2013)
Competition and Coexistence, edited by U. Sommer, B. Worm (Springer-Verlag, Berlin, 2002)
B. Karrer, M.E.J. Newman, Phys. Rev. E 84, 036106 (2011)
D. O’Sullivan, G.L.W. Perry, Spatial Simulations: Exploring Pattern and Process (John Wiley & Sons, Chichester, 2012)
D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)
B. Bollobás, Random Graphs, 2nd edn. (Cambridge UP, Cambridge, 2001)
C. Castellano, AIP Conf. Proc. 779, 114 (2005)
V. Sood, T. Antal, S. Redner, Phys. Rev. E 77, 041121 (2008)
K. Sznajd-Weron, J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000)
K. Sznajd-Weron, M. Tabiszewski, A.M. Timpanaro, Europhys. Lett. 96, 48002 (2011)
R.S. Ferreira, S.C. Ferreira, Eur. Phys. J. B 86, 462 (2013)
R. Dickman, M.M. de Oliveira, Physica A 357, 134 (2005)
S.C. Fereira, R.S. Fereira, R. Pastor-Satorras, Phys. Rev. E 83, 066113 (2011)
E.A. van Doorn, P.K. Pollett, Eur. J. Oper. Res. 230, 1 (2013)
W. de Nooy, A. Mrvar, V. Batagelj, Exploratory Social Network Analysis with Pajek, 2nd edn. (Cambridge UP, Cambridge, 2012)
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Rybak, M., Malarz, K. & Kułakowski, K. Competing contact processes in the Watts-Strogatz network. Eur. Phys. J. B 89, 145 (2016). https://doi.org/10.1140/epjb/e2016-70135-2
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DOI: https://doi.org/10.1140/epjb/e2016-70135-2