Analytic approach to co-evolving dynamics in complex networks: dissatisfied adaptive snowdrift game

We investigate the formulation of mean-field (MF) approaches for co-evolving dynamic model systems, focusing on the accuracy and validity of different schemes in closing MF equations. Within the context of a recently introduced co-evolutionary snowdrift game in which rational adaptive actions are driven by dissatisfaction in the payoff, we introduce a method to test the validity of closure schemes and analyse the shortcomings of previous schemes. A previous scheme suitable for adaptive epidemic models is shown to be invalid for the model studied here. A binomial-style closure scheme that significantly improves upon the previous schemes is introduced. Fixed-point analysis of the MF equations not only explains the numerical observed transition between a connected state with suppressed cooperation and a highly cooperative disconnected state, but also reveals a previously undetected connected state that exhibits the unusual behaviour of decreasing cooperation as the temptation for uncooperative action drops. We proposed a procedure for selecting proper initial conditions to realize the unusual state in numerical simulations. The effects of the mean number of connections that an agent carries are also studied.