Martingales and the characteristic functions of absorption time on bipartite graphs

Abstract

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

“…To review an application of Wald’s martingale to the Moran process, see [11]. To review an application of martingales for fixation time CCFs on bipartite graphs, see [41].…”

“…The EGT literature primarily studies the conditional distributions or moments of T [21,23,24,59,60]. We question whether T is an ideal quantity to represent the duration of the Moran process on a graph [41]. The graph's transition probabilities are unaffected if we eliminate time steps where the graph does not change.…”

“…Martingales are a powerful tool that can address fundamental problems in EGT and other biologically themed stochastic processes [6,11,26,27,41,[46][47][48][49]. The mutant population on an evolutionary graph may be considered as a multi-dimensional random walk between two absorbing barriers.…”

“…We will then obtain the fixation probability and the full CCFs of ‘active steps’ from that martingale [23]. Our results are the first general expressions for the full CCFs of any proxy to fixation time on a graph with more than two partitions [41]. We only require the elimination of time steps where the mutant population size does not change, and that the graph’s partitions are connected by streets.…”

Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

“…One such quantity is the duration of the process until fixation occurs [37][38][39]. For example, achieving short fixation times in combination with increasing the fixation probability does not appear to be easy.…”