Phase Transitions of the Moran Process and Algorithmic Consequences

TL;DR

This paper investigates the phase transitions and absorption times in the Moran process on graphs, establishing bounds, phase transition phenomena, and improved algorithms for fixation probability approximation.

Contribution

It provides nearly tight bounds on expected absorption time, identifies a phase transition in fixation probability, and offers an improved FPRAS for fixation probability estimation.

Findings

Expected absorption time is at most n^3 * exp(O((log log n)^3))

Phase transition in fixation probability depending on mutation fitness r

No similar phase transition occurs in directed graphs

Abstract

The Moran process is a random process 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 expected absorption time. For all epsilon > 0, we show that the expected absorption 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 absorption 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 and some basic properties of the graph are given.