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The Moran process on a random graph

Alan Frieze, Wesley Pegden

TL;DR

The paper analyzes Moran-type fixation probabilities for BD and DB dynamics on $G_{n,p}$ at the connectivity threshold, showing that fixation outcomes are largely governed by the local degree structure around the initial mutant. It develops a rigorous, high-probability framework combining Chernoff-type bounds, random-walk recurrences, and detailed case analysis to relate $p_+$ and $p_-$ and to characterize fixation probabilities in various regimes of $s$ and $|X|$. Key results include: (i) when $s>1$ and $np\gg\log n$, fixation probabilities align with a near-regular graph bias giving $\phi\to (s-1)/s$; (ii) for $s=1$, fixation probabilities reduce to a degree-based inverse-proportional formula; (iii) for $s<1$, fixation is $o(1)$; and (iv) in the critical threshold regime, the initial vertex degree (and possibly a neighbor) largely determines fixation, despite global heterogeneity. The findings illuminate how degree heterogeneity at the connectivity threshold shapes competition dynamics and provide a toolkit for analyzing fixation on sparse random graphs.

Abstract

We study the fixation probability for two versions of the Moran process on the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness $s$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex $v_0$. In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants $X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants $X$ is empty or $[n]$. The {\em fixation probability} is the probability that the process ends with $X=\emptyset$. We give asymptotically correct estimates of the fixation probability that depend on degree of $v_0$ and its neighbors.,

The Moran process on a random graph

TL;DR

The paper analyzes Moran-type fixation probabilities for BD and DB dynamics on at the connectivity threshold, showing that fixation outcomes are largely governed by the local degree structure around the initial mutant. It develops a rigorous, high-probability framework combining Chernoff-type bounds, random-walk recurrences, and detailed case analysis to relate and and to characterize fixation probabilities in various regimes of and . Key results include: (i) when and , fixation probabilities align with a near-regular graph bias giving ; (ii) for , fixation probabilities reduce to a degree-based inverse-proportional formula; (iii) for , fixation is ; and (iv) in the critical threshold regime, the initial vertex degree (and possibly a neighbor) largely determines fixation, despite global heterogeneity. The findings illuminate how degree heterogeneity at the connectivity threshold shapes competition dynamics and provide a toolkit for analyzing fixation on sparse random graphs.

Abstract

We study the fixation probability for two versions of the Moran process on the random graph at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex . In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants is empty or . In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants is empty or . The {\em fixation probability} is the probability that the process ends with . We give asymptotically correct estimates of the fixation probability that depend on degree of and its neighbors.,
Paper Structure (21 sections, 13 theorems, 86 equations)

This paper contains 21 sections, 13 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Chernoff Bounds
  4. Random Graph Properties
  5. Birth-Death
  6. The size of $(X\cup N(X))\cap S_1$
  7. $p_+$ versus $p_-$
  8. Fixation probability -- Proof of Theorem \ref{['th1']}
  9. Case analysis
  10. $s<1$
  11. $s=1$
  12. $np\gg\log n$ and $s>1$
  13. Death-Birth
  14. The size of $(X\cup N(X))\cap S_1$
  15. Bounds on $p_+,p_-$
  16. ...and 6 more sections

Key Result

Theorem 1

Given a graph $G$ and a vertex $v_0$, we let $\phi=\phi^{\mathrm{BD}}_{G,v_0,r}$ denote the fixation probability of the Birth-Death process on $G$ when the process is initialized with a mutant at $v_0$ of relative fitness $s>1$. If $G$ is a random graph sampled according to the distribution $G_{n,p} On the other hand, if $s\leq 1$ then $G$ has the property that $\phi_{G,v_0}=o(1)$ regardless of $v

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 16 more