Eur. Phys. J. B 80, 233-241 (2011)
https://doi.org/10.1140/epjb/e2011-10845-3

Analyzing phase diagrams and phase transitions in networked competing populations

¹ Jiangsu Key Laboratory of Thin Films, School of Physical Science and Technology, Soochow University, Suzhou, 215006, P.R. China
² Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, P.R. China

Corresponding author: ^a cxu@suda.edu.cn

Received: 5 November 2010
Revised: 30 January 2011
Published online: 4 March 2011

Abstract

Phase diagrams exhibiting the extent of cooperation in an evolutionary snowdrift game implemented in different networks are studied in detail. We invoke two independent payoff parameters, unlike a single payoff often used in most previous works that restricts the two payoffs to vary in a correlated way. In addition to the phase transition points when a single payoff parameter is used, phase boundaries separating homogeneous phases consisting of agents using the same strategy and a mixed phase consisting of agents using different strategies are found. Analytic expressions of the phase boundaries are obtained by invoking the ideas of the last surviving patterns and the relative alignments of the spectra of payoff values to agents using different strategies. In a Watts-Strogatz regular network, there exists a re-entrant phenomenon in which the system goes from a homogeneous phase into a mixed phase and re-enters the homogeneous phase as one of the two payoff parameters is varied. The non-trivial phase diagram accompanying this re-entrant phenomenon is quantitatively analyzed. The effects of noise and cooperation in randomly rewired Watts-Strogatz networks are also studied. The transition between a mixed phase and a homogeneous phase is identify to belong to the directed percolation universality class. The methods used in the present work are applicable to a wide range of problems in competing populations of networked agents.