Strong Bounds for Evolution in Undirected Graphs

TL;DR

This paper introduces new graph structures called selective amplifiers and suppressors that influence evolutionary dynamics, providing bounds on fixation probabilities in the generalized Moran process.

Contribution

It establishes the existence of strong selective amplifiers and suppressors, and derives tight bounds on fixation probabilities, including the Thermal Theorem, for undirected graphs.

Findings

Existence of strong selective amplifiers and suppressors.

Derived upper bounds for fixation probability.

Proved the Thermal Theorem for undirected graphs.

Abstract

This work studies the generalized Moran process, as introduced by Lieberman et al. [Nature, 433:312-316, 2005]. We introduce the parameterized notions of selective amplifiers and selective suppressors of evolution, i.e. of networks (graphs) with many "strong starts" and many "weak starts" for the mutant, respectively. We first prove the existence of strong selective amplifiers and of (quite) strong selective suppressors. Furthermore we provide strong upper bounds and almost tight lower bounds (by proving the "Thermal Theorem") for the traditional notion of fixation probability of Lieberman et al., i.e. assuming a random initial placement of the mutant.