Properties of interaction networks, structure coefficients, and benefit-to-cost ratios

In structured populations the spatial arrangement of cooperators and defectors on the interaction graph together with the structure of the graph itself determines the game dynamics and particularly whether or not fixation of cooperation (or defection) is favored. For a single cooperator (and a single defector) and a network described by a regular graph the question of fixation can be addressed by a single parameter, the structure coefficient. As this quantity is generic for any regular graph, we may call it the generic structure coefficient. For two and more cooperators (or several defectors) fixation properties can also be assigned by structure coefficients. These structure coefficients, however, depend on the arrangement of cooperators and defectors which we may interpret as a configuration of the game. Moreover, the coefficients are specific for a given interaction network modeled as regular graph, which is why we may call them specific structure coefficients. In this paper, we study how specific structure coefficients vary over interaction graphs and link the distributions obtained over different graphs to spectral properties of interaction networks. We also discuss implications for the benefit-to-cost ratios of donation games.