Emergence of a complex network structure on a Spatial Prisoner’s Dilemma

Many studies have proposed spatial game theory on network systems. Heterogeneous structures seem to contribute to population dynamics. However, few studies have addressed both dynamical population evolution and network growth events, especially incorporating individual players’ decision-making processes into the model. In this study, we considered a spatial prisoner’s dilemma (SPD) on a random network. In our model, the players were allowed to access the recent past information on themselves and neighboring players. In the “unlikely to happen” scenario, players adopted a strategy that rarely happens, which may have brought some risks to players. Moreover, the players in our model evolved their link with other players by altering their neighborhood when they received a low payoff. As a result, we found that our model spontaneously evolved as an approximate scale-free network around a critical parameter. Interestingly, hub players sometimes decreased their node degree; thus, these players are changeable in our system.Author summary: This study explores the coevolution of strategies and network structure in a spatial prisoner’s dilemma on a random network. Players adapt their strategies by occasionally adopting rare and risky behaviors that may not frequently occur but can influence outcomes. Moreover, when players receive low payoffs, they actively adjust their social links by changing neighbors, allowing the network to evolve dynamically. This interplay between strategy updates and network rewiring drives the spontaneous formation of an approximate scale-free network near a critical parameter value. Interestingly, players who become hubs in the network do not maintain constant connectivity; instead, their number of connections fluctuates over time. These results emphasize the importance of incorporating both individual decision-making processes and network evolution in models of population dynamics. Understanding such mechanisms provides valuable insights into how complex social and biological networks grow and change over time.