# Exact numerical calculation of fixation probability and time on graphs

@article{Hindersin2016ExactNC, title={Exact numerical calculation of fixation probability and time on graphs}, author={Laura Hindersin and Marius M{\"o}ller and Arne Traulsen and Benedikt Bauer}, journal={Bio Systems}, year={2016}, volume={150}, pages={ 87-91 } }

## 37 Citations

### Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

- MathematicsRoyal Society Open Science
- 2022

Evolutionary graph theory (EGT) investigates the Moran birth–death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of…

### Path to fixation of evolutionary processes in graph-structured populations

- Mathematics
- 2021

We study the spreading of a single mutant in graph-structured populations with a birth-death update rule. We use a mean-field approach and a Markov chain dynamics to investigate the effect of network…

### Martingales and the fixation probability of high-dimensional evolutionary graphs.

- MathematicsJournal of theoretical biology
- 2018

### Fixation probabilities in graph-structured populations under weak selection

- MathematicsPLoS Comput. Biol.
- 2021

This work derives an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce, and enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time.

### Spectral analysis of transient amplifiers for death–birth updating constructed from regular graphs

- MathematicsJournal of mathematical biology
- 2021

A perturbation method for identifying transient amplifiers for death–birth updating and a spectral analysis is carried out and it is shown that the graphs from which transient amplifier can be constructed share certain structural properties.

### Constructing transient amplifiers for death-Birth updating: A case study of cubic and quartic regular graphs

- MathematicsArXiv
- 2020

A perturbation methods for identifying transient amplifiers for death-Birth updating and a spectral analysis is carried out and it is shown that the graphs from which the transient amplifier can be constructed share certain structural properties.

### Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

- MathematicsCommunications Biology
- 2019

A genetic algorithm is developed to find evolutionary graphs with varying fixation probability and time and structural properties that maximize or minimize both of these parameters are identified, which allows for a first map of the universe of evolutionary graphs.

### Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs

- MathematicsPLoS Comput. Biol.
- 2020

The first known examples of transient amplifiers of selection (graphs that amplify selection for a particular range of fitness values) for the death-Birth process are uncovered and new families of “reducers of fixation” are exhibited, which decrease the fixation probability of all mutations, whether beneficial or deleterious.

### A theory of evolutionary dynamics on any complex spatial structure

- BiologybioRxiv
- 2021

This work builds network generation algorithms, evolutionary simulations and derive general analytic approximations for probabilities of fixation in populations with complex spatial structure by tuning network parameters and properties independent of each other, and shows that both a network’s degree distribution and its node mixing pattern shape the evolutionary dynamics of new mutations.

### Exact fixation probabilities for the Birth-Death and Death-Birth frequency-dependent Moran processes on the star graph

- Mathematics
- 2020

This work generalizes Broom and Rychtař's solution for the fixation probabilities of the Moran process for a structured population by allowing individuals' fitnesses to depend on the population frequency, and also by allowing a possible change in the order of reproduction and death draws.

## References

SHOWING 1-10 OF 40 REFERENCES

### Fast and asymptotic computation of the fixation probability for Moran processes on graphs

- MathematicsBiosyst.
- 2015

### Counterintuitive properties of the fixation time in network-structured populations

- Computer ScienceJournal of The Royal Society Interface
- 2014

The Moran process is studied, a discrete time birth–death process that describes the invasion of a mutant type into a population of wild-type individuals, and analytically it is shown that the time to fixation can decrease when links are removed from the network and the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant.

### A Novel Analytical Method for Evolutionary Graph Theory Problems

- Computer Science, MathematicsBiosyst.
- 2013

### Fixation probabilities for simple digraphs

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2013

The problem of finding birth–death fixation probabilities for configurations of normal and mutants on an N-vertex graph is formulated in terms of a Markov process on the 2N-dimensional state space of…

### Analytical calculation of average fixation time in evolutionary graphs.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2015

An analytical approach is introduced for exact calculation of Average Fixation Time (AFT) for two types of evolutionary graphs: cycle graph, as a highly homogeneous graph and star graph,As a highly heterogeneous graph, which uses symmetries of these graphs to calculate AFT.

### Evolutionary dynamics on graphs

- MathematicsNature
- 2005

This work determines the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population, and shows that the outcome of evolutionary games can depend entirely on the structure of the underlying graph.

### An analysis of the fixation probability of a mutant on special classes of non-directed graphs

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2008

There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005…

### Evolutionary games on graphs and the speed of the evolutionary process

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2009

This paper investigates evolutionary games with the invasion process updating rules on three simple non-directed graphs: the star, the circle and the complete graph and derives the exact solutions of the stochastic evolutionary game dynamics.

### Universality of fixation probabilities in randomly structured populations

- MathematicsScientific reports
- 2014

It is proved that for a generalization of the Erdős-Rényi model, the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability.

### Birth–death fixation probabilities for structured populations

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2013

This paper presents an adaptation of the Moran birth–death model of evolutionary processes on graphs. The present model makes use of the full population state space consisting of 2N binary-valued…