The Mixed Birth-death/death-Birth Moran Process

TL;DR

This paper introduces and analyzes a unified mixed Moran process combining birth-death and death-birth steps on graphs, providing exact formulas, probabilistic bounds, and polynomial-time approximation methods for fixation probabilities and absorption times.

Contribution

It formalizes the $-mixed Moran process, analyzes its behavior on various graph classes, and derives explicit formulas and bounds for fixation probabilities and absorption times.

Findings

At $=1/2$, fixation probability for $r=1$ is exactly $1/n$.

At $=1/2$, absorption time is $O(n^4)$ for any $r$.

For graphs with two degree values, explicit fixation formulas and bounds are provided.

Abstract

We study evolutionary dynamics on graphs in which each step consists of one birth and one death, also known as the Moran processes. There are two types of individuals: residents with fitness $1$ and mutants with fitness $r$. Two standard update rules are used in the literature. In Birth-death (Bd), a vertex is chosen to reproduce proportional to fitness, and one of its neighbors is selected uniformly at random to be replaced by the offspring. In death-Birth (dB), a vertex is chosen uniformly to die, and then one of its neighbors is chosen, proportional to fitness, to place an offspring into the vacancy. We formalize and study a unified model, the $\lambda$-mixed Moran process, in which each step is independently a Bd step with probability $\lambda \in [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $\lambda=1/2$, for any graph that the fixation probability when $r=1$ with a single mutant initially on the graph is exactly $1/n$, and also at $\lambda=1/2$ that the absorption time for any $r$ is $O_r(n^4)$. We also show results for graphs that are "almost regular," in a manner defined in the paper. We use this to show that for suitable random graphs from $G \sim G(n,p)$ and fixed $r>1$, with high probability over the choice of graph, the absorption time is $O_r(n^4)$, the fixation probability is $\Omega_r(n^{-2})$, and we can approximate the fixation probability in polynomial time. Another special case is when the graph has only two distinct degree values $\{d_1, d_2\}$ with $d_1 \leq d_2$. For those graphs, we give exact formulas for fixation probabilities when $r = 1$ and any $\lambda$, and establish an absorption time of $O_r(n^4 \alpha^4)$ for all $\lambda$, where $\alpha = d_2 / d_1$. We also provide explicit formulas for the star and cycle under any $r$ or $\lambda$.