{"ID":2837872,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.18252","arxiv_id":"2511.18252","title":"The Mixed Birth-death/death-Birth Moran Process","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 $r\u003e1$, 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 $λ$.","short_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 t...","url_abs":"https://arxiv.org/abs/2511.18252","url_pdf":"https://arxiv.org/pdf/2511.18252v3","authors":"[\"David A. Brewster\",\"Yichen Huang\",\"Michael Mitzenmacher\",\"Martin A. Nowak\"]","published":"2025-11-23T02:43:14Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"cs.DS\"]","methods":"[]","has_code":false}