In general models of games on graphs, individuals are placed on two graphs with the same set of vertices : the interaction graph and the replacement graph. A graph consists of a set of vertices and a set of edges connecting pairs of vertices. and for reviews) go beyond these two simplifying assumptions by considering finite-sized populations embedded in graphs representing geographical isolation or social networks. In spite of the importance of the replicator dynamics as a mathematical tool for investigating evolutionary dynamics, it is obvious that real populations are never infinite nor perfectly well-mixed. When populations are assumed to be infinite and well-mixed, the replicator dynamics, offers a deterministic and exact account of the evolutionary dynamics. Behavioral strategies are genetically or culturally inherited, so that the relative abundance of fitter strategies increases over time due to natural selection or social learning. Evolutionary game theory builds on the theory of games by considering populations of individuals whose success or fitness depends on the outcome of social interactions. Since the pioneering work of Maynard Smith and Price, evolutionary game theory has become a valuable tool to describe and study evolutionary dynamics when fitness is frequency-dependent. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner’s dilemma. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner’s dilemma. By combining evolutionary game theory and graph theory, “games on graphs” study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks.

Bipartite graph