# Faster Monte-Carlo Algorithms for Fixation Probability of the Moran Process on Undirected Graphs

## TL;DR

This paper introduces faster Monte-Carlo algorithms for calculating the fixation probability in the Moran process on undirected graphs, significantly improving efficiency over previous methods with proven lower bounds.

## Contribution

The authors develop polynomial-time Monte-Carlo algorithms that are at least O(n^2/log n) faster than prior algorithms for undirected graphs, with tight lower bounds on their expected steps.

## Key findings

- Algorithms are at least O(n^2/log n) faster than previous methods.
- Algorithms are polynomial time even with binary input for r.
- Lower bounds match the upper bounds asymptotically.

## Abstract

Evolutionary graph theory studies the evolutionary dynamics in a population structure given as a connected graph. Each node of the graph represents an individual of the population, and edges determine how offspring are placed. We consider the classical birth-death Moran process where there are two types of individuals, namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates the reproductive strength. The evolutionary dynamics happens as follows: in the initial step, in a population of all resident individuals a mutant is introduced, and then at each step, an individual is chosen proportional to the fitness of its type to reproduce, and the offspring replaces a neighbor uniformly at random. The process stops when all individuals are either residents or mutants. The probability that all individuals in the end are mutants is called the fixation probability. We present faster polynomial-time Monte-Carlo algorithms for finidng the fixation probability on undirected graphs. Our algorithms are always at least a factor O(n^2/log n) faster as compared to the previous algorithms, where n is the number of nodes, and is polynomial even if r is given in binary. We also present lower bounds showing that the upper bound on the expected number of effective steps we present is asymptotically tight for undirected graphs.