Absorption and fixation times for evolutionary processes on graphs

TL;DR

This work analyzes absorption and fixation times for evolutionary processes on graphs under Bernoulli and binomial proliferation, revealing two key thresholds: a critical $p_c$ that equates fixation probabilities with the Moran process, and a second threshold $p_t$ beyond which mean fixation time decreases with proliferation. The authors derive exact recurrences and closed-form expressions for absorption and fixation times on complete graphs, cycles, and stars under Bernoulli proliferation, and extend to binomial proliferation on complete graphs, using Moran-type frameworks and graphical methods. They characterize a graph-dependent G-symmetry, showing Maruyama-Kimura-type symmetries hold only for cliques and cycles, with symmetry breaking on other graphs leading to shifts in the r-value that maximizes fixation time. Using Harris’ graphical construction, they prove monotonicity of fixation time with respect to the proliferation parameter and establish the existence of a time-based phase transition $p_t$, delineating regimes where proliferation is time advantageous or disadvantageous. The findings connect absorption and fixation dynamics to cover-time concepts for simple random walks and highlight rich graph-structure effects on evolutionary processes, with implications for predicting spread in structured populations.

Abstract

In this paper, we study the absorption and fixation times for evolutionary processes on graphs, under different updating rules. While in Moran process a single neighbour is randomly chosen to be replaced, in proliferation processes other neighbours can be replaced using Bernoulli or binomial draws depending on $0 < p \leq 1$. There is a critical value $p_c$ such that the proliferation is advantageous or disadvantageous in terms of fixation probability depending on whether $p > p_c$ or $p < p_c$. We clarify the role of symmetries for computing the fixation time in Moran process. We show that the Maruyama-Kimura symmetry depend on the graph structure induced in each state, implying asymmetry for all graphs except cliques and cycles. There is a fitness value, not necessarily $1$, beyond which the fixation time decreases monotonically. We apply Harris' graphical method to prove that the fixation time decreases monotonically depending on $p$. Thus there exists another value $p_t$ for which the proliferation is advantageous or disadvantageous in terms of time. However, at the critical level $p=p_c$, the proliferation is highly advantageous when $r \to +\infty$.

Figures (11)

• Figure 1: Mean absorption and fixation times for the Moran process on all graphs of order $6$.
• Figure 2: Bernoulli proliferation on a complete graph: \ref{['fig:initialcomplete']} the initial state $S=\{1,2,3\}$, \ref{['fig:statecomplete']} the reduced state space.
• Figure 3: Bernoulli proliferation on a cycle: \ref{['fig:initialcycle']} the initial state $S=\{1\}$, \ref{['fig:statecycle']} the reduced state space.
• Figure 4: Bernoulli proliferation on a star: \ref{['fig:initialstar']} central initial state, \ref{['fig:statestar']} the reduced state space.
• Figure 5: Binomial jump proliferation on a complete graph of \ref{['fig:binomial3']} order $N=3$, \ref{['fig:binomial4']} order $N=4$.

Theorems & Definitions (11)

• Remark 1
• proof
• Remark 2
• proof
• Remark 3
• proof : Proof of Theorem \ref{['thm:decreaseMoran']}
• proof : Proof of Theorem \ref{['thm:decreasebeta']}
• proof
• proof
• proof