scholarly journals Birth–death fixation probabilities for structured populations

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120248 ◽  
Author(s):  
Burton Voorhees
Keyword(s):  
Complete Graph ◽  
Present Model ◽  
Automorphism Groups ◽  
Linear Equations ◽  
Structured Populations ◽  
Edge Weight ◽  
Simple Graphs ◽  
Fixation Probabilities ◽  
Fitness Parameter ◽  
Birth Death

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 2 N binary-valued vectors, and a Markov process on this space with a transition matrix defined by the edge weight matrix for any given graph. While the general case involves solution of 2 N – 2 linear equations, symmetry considerations substantially reduce this for graphs with large automorphism groups, and a number of simple examples are considered. A parameter called graph determinacy is introduced, measuring the extent to which the fate of any randomly chosen population state is determined. Some simple graphs that suppress or enhance selection are analysed, and comparison of several examples to the Moran process on a complete graph indicates that in some cases a graph may enhance selection relative to a complete graph for only limited values of the fitness parameter.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

scholarly journals On the Length of a Random Minimum Spanning Tree

2015 ◽  
Vol 25 (1) ◽  
pp. 89-107 ◽  
Author(s):  
COLIN COOPER ◽  
ALAN FRIEZE ◽  
NATE INCE ◽  
SVANTE JANSON ◽  
JOEL SPENCER
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Expected Value ◽  
Edge Weight ◽  
Formula Group ◽  
Image Position

We study the expected value of the lengthLnof the minimum spanning tree of the complete graphKnwhen each edgeeis given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$wherec1,c2are explicitly defined constants.

scholarly journals Evolution of cooperation in an epithelium

2019 ◽  
Vol 16 (152) ◽  
pp. 20180918 ◽  
Author(s):  
Jessie Renton ◽  
Karen M. Page
Keyword(s):  
Social Interactions ◽  
Voronoi Tessellation ◽  
Evolutionary Game ◽  
Cellular Level ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Animal Kingdom ◽  
Population Structures ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

scholarly journals Fixation probabilities for simple digraphs

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120676 ◽  
Author(s):  
Burton Voorhees ◽  
Alex Murray
Keyword(s):  
Complete Bipartite Graph ◽  
Directed Graphs ◽  
Closed Form Solution ◽  
Form Solution ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Fixation Probability ◽  
Single Vertex ◽  
Fixation Probabilities ◽  
Birth Death

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 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the single-vertex fixation probabilities. This also yields a closed-form solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r >1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.

scholarly journals Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

2018 ◽  
Author(s):  
David Hathcock ◽  
Steven H. Strogatz
Keyword(s):  
Complete Graph ◽  
Time Distribution ◽  
Evolutionary Dynamics ◽  
Structured Populations ◽  
Relative Fitness ◽  
Death Process ◽  
Fixation Time ◽  
Large Network ◽  
Moran Process ◽  
Fitness Dependence

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of mutant and non-mutant individuals. Recent studies have shown that fixation times, which determine the rate of evolution, often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, each of which admits an exact solution in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a weighted convolution of two Gumbel distributions, with a weight depending on the relative fitness. When fitness is neutral, however, the Moran process has a highly skewed fixation-time distribution on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model, times to partial fixation, and evolution on random networks are discussed.

scholarly journals Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

2016 ◽  
Author(s):  
F. Débarre
Keyword(s):  
Social Behavior ◽  
Life Cycles ◽  
Structured Populations ◽  
Altruistic Behavior ◽  
Common Life ◽  
Expected Frequency ◽  
Phenotypic Differences ◽  
Population Structures ◽  
Birth Death

AbstractThe theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. Here, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider populations of fixed size N, structured such that in the absence of selection, all individuals have the same probability of reproducing or dying (neutral reproductive values are the all same). Two types of individuals, A and B, corresponding to two types of social behavior, are competing; the fidelity of strategy transmission from parent to offspring is tuned by a parameter μ. Social interactions have a direct effect on individual fecundities. Under the assumption of small phenotypic differences (implyingweak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate our results with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures (graph-structured populations). Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. We also show that computing the expected frequency of altruists on a regular graph may require knowing more than just the graph’s size and degree.

scholarly journals On a problem of Ahlswede and Katona

2009 ◽  
Vol 46 (3) ◽  
pp. 423-435
Author(s):  
Stephan Wagner ◽  
Hua Wang
Keyword(s):  
Complete Graph ◽  
Simple Graphs

Let p ( G ) denote the number of pairs of adjacent edges in a graph G . Ahlswede and Katona considered the problem of maximizing p ( G ) over all simple graphs with a given number n of vertices and a given number N of edges. They showed that p ( G ) is either maximized by a quasi-complete graph or by a quasi-star. They also studied the range of N (depending on n ) for which the quasi-complete graph is superior to the quasi-star (and vice versa) and formulated two questions on distributions in this context. This paper is devoted to the solution of these problems.

scholarly journals On the Automorphism Groups of Quasiprimitive Almost Simple Graphs

1999 ◽  
Vol 222 (1) ◽  
pp. 271-283 ◽  
Author(s):  
Xin Gui Fang ◽  
George Havas ◽  
Cheryl E. Praeger
Keyword(s):  
Automorphism Groups ◽  
Simple Graphs
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