There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. Nature 433, 312-316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations.

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Citation: Pattni, K., Broom, M., Rychtar, J. & Silvers, L. J. (2015). Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471, e2182. doi: 10.1098/rspa.2015 This is the accepted version of the paper.This version of the publication may differ from the final published version. Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population, is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics. Permanent

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractRecently, models of evolution have begun to incorporate structured populations, including spatial structure, through the modelling of evolutionary processes on graphs (evolutionary graph theory). One limitation of this otherwise quite general framework is that interactions are restricted to pairwise ones, through the edges connecting pairs of individuals. Yet many animal interactions can involve many players, and theoretical models also describe such multi-player interactions. We shall discuss a more general modelling framework of interactions of structured populations with the focus on competition between territorial animals, where each animal or animal group has a "home range" which overlaps with a number of others, and interactions between various group sizes are possible. Depending upon the behaviour concerned we can embed the results of different evolutionary games within our structure, as occurs for pairwise games such as the Prisoner's Dilemma * Corresponding author. Email addresses: mark.broom@city.ac.uk (Mark Broom), rychtar@uncg.edu (Jan Rychtář) Preprint submitted to Journal of Theoretical Biology February 23, 2012 or the Hawk-Dove game on graphs. We discuss some examples together with some important differences between this approach and evolutionary graph theory.

In this paper, we investigate evolutionary games with the invasion process updating rules on three simple non-directed graphs: the star, the circle and the complete graph. Here, we present an analytical approach and derive the exact solutions of the stochastic evolutionary game dynamics. We present formulae for the fixation probability and also for the speed of the evolutionary process, namely for the mean time to absorption (either mutant fixation or extinction) and then the mean time to mutant fixation. Through numerical examples, we compare the different impact of the population size and the fitness of each type of individual on the above quantities on the three different structures. We do this comparison in two specific cases. Firstly, we consider the case where mutants have fixed fitness r and resident individuals have fitness 1. Then, we consider the case where the fitness is not constant but depends on games played among the individuals, and we introduce a 'hawk-dove' game as an example.