A Novel Analytical Method for Evolutionary Graph Theory Problems

TL;DR

This paper introduces a new deterministic analytical framework for calculating fixation probabilities in evolutionary graph theory, outperforming simulations in speed and providing insights into complex biological and neurological systems.

Contribution

The paper presents a novel deterministic method for fixation probability computation in directed, weighted graphs, extending analysis capabilities beyond existing simulation-based approaches.

Findings

Framework computes fixation probabilities efficiently on various graphs.

Method outperforms Monte Carlo simulations in speed by several orders of magnitude.

Provides insights into synaptic competition in neurology.

Abstract

Evolutionary graph theory studies the evolutionary dynamics of populations structured on graphs. A central problem is determining the probability that a small number of mutants overtake a population. Currently, Monte Carlo simulations are used for estimating such fixation probabilities on general directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic framework for computing fixation probabilities for strongly connected, directed, weighted evolutionary graphs under neutral drift. We show how this framework can also be used to calculate the expected number of mutants at a given time step (even if we relax the assumption that the graph is strongly connected), how it can extend to other related models (e.g. voter model), how our framework can provide non-trivial bounds for fixation probability in the case of an advantageous mutant, and how it can be used to find a non-trivial lower bound on the mean time to fixation. We provide various experimental results determining fixation probabilities and expected number of mutants on different graphs. Among these, we show that our method consistently outperforms Monte Carlo simulations in speed by several orders of magnitude. Finally we show how our approach can provide insight into synaptic competition in neurology.