The Mixed Birth-death/death-Birth Moran Process

TL;DR

We introduce the λ-mixed Moran process to interpolate between Birth-death and death-Birth updates on graphs and analyze fixation probability and absorption time for undirected graphs. Using a potential-function drift framework, we obtain bounds and exact results across regimes, including: (i) λ=1/2 with $fp$ degenerating to $|S_0|/n$ when $r=1$ and $O_r(n^4)$ absorption for general $r$; (ii) almost-regular and random graphs yielding $O_r(n^4)$ absorption and $Ω_r(n^{-2})$ fixation with an FPRAS; (iii) explicit formulas for bidegreed graphs, especially stars and cycles; and (iv) exact formulas for cycles and stars under arbitrary $r$ and $λ$. Our results show that the mixed update dynamics can significantly differ from the pure Bd or dB cases, including non-monotonic dependence on λ, and provide a suite of techniques and special-case analyses underpinning efficient approximation of fixation probabilities. The work paves the way for broader investigations into mixed-update evolutionary dynamics on diverse graph classes and motivates future study of directed/weighted graphs and amplifier/suppressor structures under the λ-mixed framework.

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 $λ\in [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $λ=1/2$, 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 $λ$.

Figures (3)

• Figure 1: Examples of Bidegreed Graphs.
• Figure 2: Fixation probabilities for a cycle on $n=100$ vertices starting from a single mutant. Fixation probabilities are computed using \ref{['eq: cycle value']}. The two parameters are $r$ and $\lambda$, with $\lambda$ encoded by color. When $r<1$, the fixation probabilities are very small (but not actually 0).
• Figure 3: Fixation probabilities for a star on $10$ vertices starting from a single initial mutant, computed using the results from \ref{['section: star']}. The two parameters are $r$ and $\lambda$, with $\lambda$ encoded by color. When the initial mutant is at the center, larger $\lambda$ (more Bd) consistently yields a smaller fixation probability. When starting from a leaf, there appears to be a phase transition around $r=1$ about whether greater Bd or dB gives a higher fixation probability.

Theorems & Definitions (42)

• Definition 2.1: Mixed Moran fixation
• Remark 2.2
• Theorem 2.3: Additivity
• Theorem 2.4
• Theorem 2.5: Absorption time, positive drift
• Theorem 2.6: Fixation probability, lower bound
• Theorem 2.7: FPRAS
• Theorem 3.1
• proof
• Example 3.2