The Moran process is a randomised algorithm that models the spread of genetic mutations through graphs. If the graph is connected, the process eventually reaches "fixation", where every vertex is a mutant, or "extinction", where no vertex is a mutant. Our main result is an almost-tight bound on the expected running time of the algorithm. For all epsilon > 0, we show that the expected running time on an n-vertex graph is o(n^(3+epsilon)). In fact, we show that it is at most n^3 * exp(O((log log n)^3)) and that there is a family of graphs where it is Omega(n^3). In the course of proving our main result, we also establish a phase transition in the probability of fixation, depending on the fitness parameter r of the mutation. We show that no similar phase transition occurs for digraphs, where it is already known that the expected running time can also be exponential. Finally, we give an improved FPRAS for approximating the probability of fixation. Its running time is independent of the size of the graph when the maximum degree is bounded.
@cite_3: © 2016 Wiley Periodicals, Inc. The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is O(nlogn) and O(n2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.
Prior to Theorem , it was known that for all @math -vertex graphs @math , @math is increasing in @math @cite_3 and that @math [Lemma 1] DGMRSS2014 . Thus @math for all @math , and @math for all @math . To our knowledge, these were the best previously known bounds that apply to all graphs, and the only known bound for @math . When @math , given a fixed initial mutant @math , Mertzios and Spirakis [Theorem 4] MS2013 showed that fixation occurs with probability at least @math , and a result of Giakkoupis [Lemma 6] Gia2016 implies that fixation occurs with probability at least @math . These results both imply better bounds than @math in the case where @math is sparse or has high minimum degree.