Two results on evolutionary processes on general non-directed graphs

The paper Broom & Rychtář (2008) analytically investigated the probability for mutants to fixate in an otherwise uniform population on two types of heterogeneous graphs (lines and stars) by evolutionary dynamics. The main motivation for concentrating on those two types of graphs only was the potentially exponential size of the system of linear equations (see equation (1.1) below) yielding the fixation probability on general heterogeneous graphs. The size of the system was given by formula (4.1) from Broom & Rychtář (2008). It turns out that formula (4.1) is in fact only a lower bound for the size of the system and in this paper we correct this by deriving a formula for the exact size of the system (1.1). We also solve the system (1.1) for general heterogeneous graphs in the case of random drift. Let G = (V ,E) be an undirected graph, where V is the set of vertices and E is the set of edges. We assume that the graph is finite, connected and simple, i.e. no vertex is connected to itself and there are no parallel edges. The graph structure is represented by a matrix W = (wij), where