Approximating Fixation Probabilities in the Generalized Moran Process

We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312–316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned ‘fitness’ value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness $r>0$ placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when $r\geq 1$) and of extinction (for all $r>0$).

💡 Research Summary

The paper studies the generalized Moran process introduced by Lieberman, Hauert, and Nowak, where individuals occupy the vertices of a finite, connected, undirected graph. At each discrete time step a vertex is selected with probability proportional to its fitness; the selected individual reproduces by copying itself onto a uniformly random neighbor, replacing the resident there. Initially there is a single mutant of fitness r > 0 placed uniformly at random, while all other vertices host residents of fitness 1. The process inevitably reaches one of two absorbing states: fixation (all vertices become mutants) or extinction (no mutants remain).

Exact computation of the fixation probability f_{G,r} requires solving a linear system of size 2^{|V|}, which is infeasible for all but the smallest graphs. Prior work therefore focused on highly symmetric families (paths, cycles, stars, complete graphs) or on empirical studies. This paper overcomes that limitation by proving that, with high probability, the number of steps needed to reach absorption is bounded by a polynomial in the number of vertices n, regardless of the graph’s structure.

The authors first establish that when r = 1 the fixation probability is exactly 1/n for any connected graph. The proof uses a “color‑propagation” argument: assign each vertex a distinct color, run the neutral Moran process, and observe that with probability 1 a single color eventually dominates; the chance that the initial mutant’s color wins is therefore 1/n. From this they deduce that for r > 1 the fixation probability is strictly larger than 1/n, while for r < 1 it can be exponentially small (e.g., on the complete graph K_n).

The technical heart of the paper is the introduction of a potential function φ(S) = ∑_{x∈S} 1/deg(x) for a mutant set S. φ satisfies 1 < φ(G) < n and φ({x}) ≤ 1 for any vertex x. Lemma 5 shows that the expected change of φ in one step is positive when r > 1 and negative when r < 1, with magnitude at least |r−1|·n^{−3}. This monotonicity enables the use of martingale techniques. Theorem 6 (a standard super‑martingale stopping‑time bound) together with Lemma 5 yields explicit upper bounds on the expected absorption time τ: for r < 1, E