Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1501.06342

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

Fernando Alcalde Cuesta·Pablo González Sequeiros·Álvaro Lozano Rojo
View original source ↗Read source document ↗Export metadata ↓

Abstract

Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogenous case. One of the most important models has been proposed by Lieberman, Hauert and Nowak, adapting to a weighted directed graph the classical process described by Moran. The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this paper, we shall use this idea to compute asymptotically the average fixation probability for complete bipartite graphs. To this end, we firstly review some recent results on evolutionary dynamics on graphs trying to clarify some points. We also revisit the 'Star Theorem' proved by Lieberman, Hauert and Nowak for the star graphs. Theoretically, EMC techniques allow fast computation of the fixation probability, but in practice this is not always true. Thus, in the last part of the paper, we compare this algorithm with the standard Monte Carlo method for some kind of complex networks.

Explore related subjects

q-bio.PE

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fernando Alcalde Cuesta, Pablo González Sequeiros, Álvaro Lozano Rojo. 2015-02-11. Fast and asymptotic computation of the fixation probability for Moran processes on graphs. https://doi.org/10.1016/j.biosystems.2015.01.007

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Phase transitions in microbial lineage trees

Microbial populations exhibit high cell-to-cell variability, which fundamentally shapes population behavior. A striking consequence is the existence of phase transitions, where small genetic or environmental changes trigger abrupt shifts in population dynamics. While biological phase transitions have often been proposed, connecting observed behavior to the underlying physics has remained challenging. We combine population genetics with statistical physics to show how phase transitions arise naturally in microbial populations. We highlight the existence of a first-order transition in a model of bacterial plasmid engineering and find a strict lower bound on the number of plasmids that can be stably maintained in a population.

q-bio.PE↗
arXiv2026

Phylogenetic Inference and the Stickiness of Fréchet Means, via Precise Asymptotics of an Embedded Random Walk

A well-known phenomenon in statistical analyses of populations of phylogenetic trees in the Billera-Holmes-Vogtmann space is that the topology of the Fréchet mean tree can contain multifurcations (i.e., internal nodes with more than two children), which raises the practical question of whether this reflects a population-level branching structure (hard polytomy) or merely sampling variability in the data (soft polytomy). This is an instance of the more general phenomenon of "stickiness" in non-Euclidean statistics, whereby the sample Fréchet mean in certain non-positively curved stratified spaces becomes permanently trapped in a lower-dimensional stratum. In this work, we identify a particular multidimensional random walk embedded within the Fréchet mean process, and we show that the time at which stickiness occurs is determined by the largest last-passage time above zero of the coordinates of this random walk. Using this representation, we develop a fully nonparametric procedure for estimating the probability that trifurcations in a sample Fréchet mean tree will bifurcate at some future time if more observations are collected. Lastly, we apply our methodology to a problem in phylogenetics where we consider whether an observed trifurcation in the species tree of primates, glires, and tree shrews is genuinely trifurcated at the population level.

q-bio.PE↗
arXiv2026

Coexistence coalitions in propagule disperser quasi-communities

Many natural ecosystems harbor large numbers of coexisting species competing for far fewer distinct resources, in apparent defiance of the competitive exclusion principle. Various mechanisms have been proposed to explain this apparent paradox, often pertaining to organisms with a two-stage sessile--propagule life cycle. Here we develop a stochastic model class for such propagule disperser communities that combines competition--colonization trade-offs, spatial heterogeneity, demographic stochasticity, as well as inherited trait variation, and recover several classical models as special or limiting cases. Using bifurcation analysis, we classify equilibrium coalitions near the extinction threshold and give sufficient conditions for their realization by macroscopic equilibria away from the threshold, bypassing the costly numerical computation of the actual equilibrium states. Illustrative examples examine the resulting trait distributions and coalition patterns, demonstrating the interactive effects of different coexistence mechanisms.

q-bio.PE↗