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

2008 ◽  
Vol 464 (2098) ◽  
pp. 2609-2627 ◽  
Author(s):  
M Broom ◽  
J Rychtář
Keyword(s):  
Evolutionary Dynamics ◽  
Directed Graphs ◽  
Homogeneous Structure ◽  
Fixation Probability ◽  
Weighted Graphs ◽  
Simple Graphs ◽  
Single Mutant ◽  
Fixation Probabilities ◽  
Similar Solutions ◽  
Initial Starting

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 Nature 433 , 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations.

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 Fixation Probabilities of Evolutionary Graphs Based on the Positions of New Appearing Mutants

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Pei-ai Zhang
Keyword(s):  
Markov Chain ◽  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Chain Process ◽  
Fixation Probability ◽  
Spatial Structures ◽  
Single Level ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory

Evolutionary graph theory is a nice measure to implement evolutionary dynamics on spatial structures of populations. To calculate the fixation probability is usually regarded as a Markov chain process, which is affected by the number of the individuals, the fitness of the mutant, the game strategy, and the structure of the population. However the position of the new mutant is important to its fixation probability. Here the position of the new mutant is laid emphasis on. The method is put forward to calculate the fixation probability of an evolutionary graph (EG) of single level. Then for a class of bilevel EGs, their fixation probabilities are calculated and some propositions are discussed. The conclusion is obtained showing that the bilevel EG is more stable than the corresponding one-rooted EG.

scholarly journals Fixation probabilities in network structured meta-populations

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sedigheh Yagoobi ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Evolutionary Ecology ◽  
Research Topic ◽  
Fixation Probability ◽  
Popular Approach ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Selection For ◽  
Carry Over ◽  
Regular Networks

AbstractThe effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

scholarly journals Counterintuitive properties of the fixation time in network-structured populations

2014 ◽  
Vol 11 (99) ◽  
pp. 20140606 ◽  
Author(s):  
Laura Hindersin ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Transition Probabilities ◽  
Structured Populations ◽  
Death Process ◽  
Fixation Time ◽  
Fixation Probability ◽  
Mutant Type ◽  
Starting Conditions ◽  
Fixation Probabilities ◽  
Regular Networks

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.

scholarly journals Fixation probabilities in network structured meta-populations

2021 ◽  
Author(s):  
Sedigheh Yagoobi ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Evolutionary Ecology ◽  
Research Topic ◽  
Fixation Probability ◽  
Popular Approach ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Selection For ◽  
Carry Over ◽  
Regular Networks

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two-patch structure with unequal population size suppresses selection for low migration probabilities.

scholarly journals Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

2021 ◽  
Vol 182 (3) ◽  
pp. 219-242
Author(s):  
Mostafa Haghir Chehreghani ◽  
Albert Bifet ◽  
Talel Abdessalem
Keyword(s):  
Betweenness Centrality ◽  
Directed Graphs ◽  
Randomized Algorithm ◽  
Exact Algorithm ◽  
Directed Network ◽  
Weighted Graphs ◽  
Single Vertex ◽  
Small Constant ◽  
Positive Weights ◽  
Real World Datasets

Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network G and a vertex r ∈ V (G), we propose an exact algorithm to compute betweenness score of r. Our algorithm pre-computes a set ℛ𝒱(r), which is used to prune a huge amount of computations that do not contribute to the betweenness score of r. Time complexity of our algorithm depends on |ℛ𝒱(r)| and it is respectively Θ(|ℛ𝒱(r)| · |E(G)|) and Θ(|ℛ𝒱(r)| · |E(G)| + |ℛ𝒱(r)| · |V(G)| log |V(G)|) for unweighted graphs and weighted graphs with positive weights. |ℛ𝒱(r)| is bounded from above by |V(G)| – 1 and in most cases, it is a small constant. Then, for the cases where ℛ𝒱(r) is large, we present a simple randomized algorithm that samples from ℛ𝒱(r) and performs computations for only the sampled elements. We show that this algorithm provides an (ɛ, δ)-approximation to the betweenness score of r. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

Virtual-Reality graph visualization based on Fruchterman-Reingold using Unity and SteamVR

2021 ◽  
pp. 147387162110603
Author(s):  
Gerd Kortemeyer
Keyword(s):  
Virtual Reality ◽  
Directed Graphs ◽  
Dynamic Visualization ◽  
Graph Visualization ◽  
Weighted Graphs ◽  
Virtual Reality System ◽  
Newtonian Physics ◽  
Dynamic Extrusion ◽  
Relationship Of ◽  
The Relationship

The paper describes a method for the immersive, dynamic visualization of undirected, weighted graphs. Using the Fruchterman-Reingold method, force-directed graphs are drawn in a Virtual-Reality system. The user can walk through the data, as well as move vertices using controllers, while the network display rearranges in realtime according to Newtonian physics. In addition to the physics behind the employed method, the paper explains the most pertinent computational mechanisms for its implementation, using Unity, SteamVR, and a Virtual-Reality system such as HTC Vive (the source package is made available for download). It was found that the method allows for intuitive exploration of graphs with on the order of [Formula: see text] vertices, and that dynamic extrusion of vertices and realtime readjustment of the network structure allows for developing an intuitive understanding of the relationship of a vertex to the remainder of the network. Based on this observation, possible future developments are suggested.

scholarly journals Evolutionary Dynamics Do Not Motivate a Single-Mutant Theory of Human Language

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Bart de Boer ◽  
Bill Thompson ◽  
Andrea Ravignani ◽  
Cedric Boeckx
Keyword(s):  
Evolutionary Dynamics ◽  
Human Language ◽  
Single Mutant

scholarly journals Strategies for increasing fixation probabilities of recessive mutations

1991 ◽  
Vol 58 (2) ◽  
pp. 129-138 ◽  
Author(s):  
A. Caballero ◽  
P. D. Keightley ◽  
W. G. Hill
Keyword(s):  
Breeding Systems ◽  
Fixation Probability ◽  
Selection Intensity ◽  
Gene Effects ◽  
Recessive Mutations ◽  
Sib Mating ◽  
Population Structures ◽  
Recessive Genes ◽  
Fixation Probabilities ◽  
Close Inbreeding

SummaryFixation probabilities and mean times to fixation of new mutant alleles in an isogenic population having an effect on a quantitative trait under truncation selection were computed using stochastic simulation. A variety of population structures and breeding systems were studied in order to find an optimal design for maximizing the fixation probability for recessive genes without impairing that for non-recessives or delaying times to fixation. Circular mating or cycles with repeated generations of close inbreeding alternating with combination of the families proved to be very inefficient. The most successful scheme found, considering fixation probabilities and times to fixation jointly, was to practise individual selection and mate full sibs whenever possible, otherwise mate at random. The benefit was directly proportional to the number of full-sib matings performed, which, in turn, almost exclusively depended on the number of selected individuals with very little effect of selection intensity or magnitude of gene effects. Fixation rates could be well approximated by diffusion methods. When selection was practised in only one sex and, therefore, the proportion of full-sib matings could be varied from zero to one, maximizing the amount of full-sib mating was found to maximize fixation probability, at least for single mutants.

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