The Mixed Birth-death/death-Birth Moran Process

2025-11-22

AI Summary

This work addresses the modeling of evolutionary dynamics on graphs under hybrid update mechanisms. We propose the λ-mixed Moran process: at each step, the Birth–death (Bd) rule is applied with probability λ and the death–Birth (dB) rule with probability 1−λ. This constitutes the first formal unification of these two canonical update rules within a single stochastic framework. Theoretically, we prove that for λ = 1/2, the neutral fixation probability equals 1/n on any graph, and derive an upper bound on absorption time; we obtain exact analytical expressions for nearly regular and bidegree graphs; and for Erdős–Rényi random graphs G(n,p), we establish an Oᵣ(n⁴) absorption time upper bound and an Ωᵣ(n⁻²) lower bound on fixation probability. Algorithmically, we design a polynomial-time approximation algorithm. Notably, we derive closed-form solutions for star and cycle graphs under arbitrary selection strength r > 0 and mixing parameter λ.

Technology Category

• Machine Learning: Evolutionary Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

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 $λ$-mixed Moran process, in which each step is independently a Bd step with probability $λin [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $λ=1/2$, 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 $λ=1/2$ 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 $Ω_r(n^{-2})$, 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 $λ$, and establish an absorption time of $O_r(n^4 α^4)$ for all $λ$, where $α= d_2 / d_1$. We also provide explicit formulas for the star and cycle under any $r$ or $λ$.

Problem

• Modeling evolutionary dynamics on graphs using mixed Moran processes
• Analyzing fixation probabilities and absorption times for mutants
• Studying mixed update rules on regular and structured graphs

Innovation

• Unified model combining Birth-death and death-Birth processes
• Analyzing fixation probabilities on undirected connected graphs
• Establishing absorption time bounds for various graph structures