Figure 1.

Evolutionary games on graphs (or social networks) are characterized by an interaction graph and a replacement graph. The interaction graph specifies who is interacting with whom in terms of the evolutionary game that is under consideration. The replacement graph specifies the local neighbourhood for updating the strategies (i.e. who competes with whom in the selection process). The simplest geometry is that both the replacement graph and the interaction graph are given by the same cycle, where each individual has exactly two neighbours and there is no boundary. This one-dimensional geometry allows an exact calculation which can be compared with approximative methods for other geometries.