scholarly journals Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with N ≤ 14 Players

Games ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 12
Author(s):  
Hendrik Richter
Keyword(s):  
Regular Graph ◽  
Good Predictor ◽  
Computational Approach ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Regular Graphs ◽  
Weak Selection ◽  
Structure Coefficient ◽  
Low Degree ◽  
Emergence Of Cooperation

We study the emergence of cooperation in structured populations with any arrangement of cooperators and defectors on the evolutionary graph. In a computational approach using structure coefficients defined for configurations describing such arrangements of any number of mutants, we provide results for weak selection to favor cooperation over defection on any regular graph with N ≤ 14 vertices. Furthermore, the properties of graphs that particularly promote cooperation are analyzed. It is shown that the number of graph cycles of a certain length is a good predictor for the values of the structure coefficient, and thus a tendency to favor cooperation. Another property of particularly cooperation-promoting regular graphs with a low degree is that they are structured to have blocks with clusters of mutants that are connected by cut vertices and/or hinge vertices.

scholarly journals Evolutionary dynamics with game transitions

2019 ◽  
Vol 116 (51) ◽  
pp. 25398-25404 ◽  
Author(s):  
Qi Su ◽  
Alex McAvoy ◽  
Long Wang ◽  
Martin A. Nowak
Keyword(s):  
Strong Influence ◽  
Evolutionary Dynamics ◽  
Simple Game ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Altruistic Behavior ◽  
Weak Selection ◽  
Time Step ◽  
Mutual Cooperation ◽  
Connected Populations

The environment has a strong influence on a population’s evolutionary dynamics. Driven by both intrinsic and external factors, the environment is subject to continual change in nature. To capture an ever-changing environment, we consider a model of evolutionary dynamics with game transitions, where individuals’ behaviors together with the games that they play in one time step influence the games to be played in the next time step. Within this model, we study the evolution of cooperation in structured populations and find a simple rule: Weak selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, b, to the corresponding cost, c, exceedsk−k′, where k is the average number of neighbors of an individual andk′captures the effects of the game transitions. Even if cooperation cannot be favored in each individual game, allowing for a transition to a relatively valuable game after mutual cooperation and to a less valuable game after defection can result in a favorable outcome for cooperation. In particular, small variations in different games being played can promote cooperation markedly. Our results suggest that simple game transitions can serve as a mechanism for supporting prosocial behaviors in highly connected populations.

scholarly journals Evolutionary games of multiplayer cooperation on graphs

2016 ◽  
Author(s):  
Jorge Peña ◽  
Bin Wu ◽  
Jordi Arranz ◽  
Arne Traulsen
Keyword(s):  
Spatial Structure ◽  
Computer Simulations ◽  
Evolutionary Games ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Pair Approximation ◽  
Regular Graphs ◽  
Multiplayer Games ◽  
Cooperative Interactions ◽  
Analytical Approximations

AbstractThere has been much interest in studying evolutionary games in structured populations, often modelled as graphs. However, most analytical results so far have only been obtained for two-player or linear games, while the study of more complex multiplayer games has been usually tackled by computer simulations. Here we investigate evolutionary multiplayer games on graphs updated with a Moran death-Birth process. For cycles, we obtain an exact analytical condition for cooperation to be favored by natural selection, given in terms of the payoffs of the game and a set of structure coefficients. For regular graphs of degree three and larger, we estimate this condition using a combination of pair approximation and diffusion approximation. For a large class of cooperation games, our approximations suggest that graph-structured populations are stronger promoters of cooperation than populations lacking spatial structure. Computer simulations validate our analytical approximations for random regular graphs and cycles, but show systematic differences for graphs with many loops such as lattices. In particular, our simulation results show that these kinds of graphs can even lead to more stringent conditions for the evolution of cooperation than well-mixed populations. Overall, we provide evidence suggesting that the complexity arising from many-player interactions and spatial structure can be captured by pair approximation in the case of random graphs, but that it need to be handled with care for graphs with high clustering.Author SummaryCooperation can be defined as the act of providing fitness benefits to other individuals, often at a personal cost. When interactions occur mainly with neighbors, assortment of strategies can favor cooperation but local competition can undermine it. Previous research has shown that a single coefficient can capture this trade-off when cooperative interactions take place between two players. More complicated, but also more realistic models of cooperative interactions involving multiple players instead require several such coefficients, making it difficult to assess the effects of population structure. Here, we obtain analytical approximations for the coefficients of multiplayer games in graph-structured populations. Computer simulations show that, for particular instances of multiplayer games, these approximate coefficients predict the condition for cooperation to be promoted in random graphs well, but fail to do so in graphs with more structure, such as lattices. Our work extends and generalizes established results on the evolution of cooperation on graphs, but also highlights the importance of explicitly taking into account higher-order statistical associations in order to assess the evolutionary dynamics of cooperation in spatially structured populations.

The Major Transitions in Evolution

1997 ◽  
Author(s):  
John Maynard Smith ◽  
Eors Szathmary
Keyword(s):  
Full Range ◽  
Evolution Of Cooperation ◽  
Major Transitions ◽  
Insect Societies ◽  
Wide Range ◽  
Major Transition ◽  
Life Itself ◽  
History Of ◽  
Emergence Of Cooperation ◽  
Multicellular Organisms

Over the history of life there have been several major changes in the way genetic information is organized and transmitted from one generation to the next. These transitions include the origin of life itself, the first eukaryotic cells, reproduction by sexual means, the appearance of multicellular plants and animals, the emergence of cooperation and of animal societies, and the unique language ability of humans. This ambitious book provides the first unified discussion of the full range of these transitions. The authors highlight the similarities between different transitions--between the union of replicating molecules to form chromosomes and of cells to form multicellular organisms, for example--and show how understanding one transition sheds light on others. They trace a common theme throughout the history of evolution: after a major transition some entities lose the ability to replicate independently, becoming able to reproduce only as part of a larger whole. The authors investigate this pattern and why selection between entities at a lower level does not disrupt selection at more complex levels. Their explanation encompasses a compelling theory of the evolution of cooperation at all levels of complexity. Engagingly written and filled with numerous illustrations, this book can be read with enjoyment by anyone with an undergraduate training in biology. It is ideal for advanced discussion groups on evolution and includes accessible discussions of a wide range of topics, from molecular biology and linguistics to insect societies.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

scholarly journals Ordering structured populations in multiplayer cooperation games

2015 ◽  
Author(s):  
Jorge Peña ◽  
Bin Wu ◽  
Arne Traulsen
Keyword(s):  
Population Structure ◽  
Spatial Structure ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Multiplayer Games ◽  
Spatial Games ◽  
Population Structures ◽  
Single Structure ◽  
Containment Order ◽  
Update Rules

AbstractSpatial structure greatly affects the evolution of cooperation. While in two-player games the condition for cooperation to evolve depends on a single structure coefficient, in multiplayer games the condition might depend on several structure coefficients, making it difficult to compare different population structures. We propose a solution to this issue by introducing two simple ways of ordering population structures: the containment order and the volume order. If population structure 𝒮1 is greater than population structure 𝒮2 in the containment or the volume order, then 𝒮1 can be considered a stronger promoter of cooperation. We provide conditions for establishing the containment order, give general results on the volume order, and illustrate our theory by comparing different models of spatial games and associated update rules. Our results hold for a large class of population structures and can be easily applied to specific cases once the structure coefficients have been calculated or estimated.

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals On Regular Factors in Regular Graphs with Small Radius

10.37236/1760 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Arne Hoffmann ◽  
Lutz Volkmann
Keyword(s):  
Regular Graph ◽  
Small Radius ◽  
Regular Graphs ◽  
High Eccentricity ◽  
K Factor

In this note we examine the connection between vertices of high eccentricity and the existence of $k$-factors in regular graphs. This leads to new results in the case that the radius of the graph is small ($\leq 3$), namely that a $d$-regular graph $G$ has all $k$-factors, for $k|V(G)|$ even and $k\le d$, if it has at most $2d+2$ vertices of eccentricity $>3$. In particular, each regular graph $G$ of diameter $\leq3$ has every $k$-factor, for $k|V(G)|$ even and $k\le d$.

scholarly journals Effects of aspiration-induced adaptation and migration on the evolution of cooperation

2014 ◽  
Vol 25 (07) ◽  
pp. 1450025 ◽  
Author(s):  
Yu Peng ◽  
Xu-Wen Wang ◽  
Qian Lu ◽  
Qing-Ke Zeng ◽  
Bing-Hong Wang
Keyword(s):  
Data Envelopment Analysis ◽  
Prospect Theory ◽  
Evolution Of Cooperation ◽  
Square Lattice ◽  
Data Envelopment ◽  
Aspiration Level ◽  
Aspiration Levels ◽  
Emergence Of Cooperation ◽  
And Migration ◽  
The Individual

In the light of the prospect theory (PT), we study the prisoner's dilemma game (PDG) on square lattice by integrating the deterministic and Data envelopment analysis (DEA) efficient rule into adaptive rules: the individual will change evolutionary rule and migrate if its payoff is lower than their aspiration levels. Whether the individual choose to change the evolutionary rule and migrate is determined by the relation between its payoff and aspiration level. The results show that the cooperation frequency can hold unchange with the increasing of temptation to defect. The individual chooses to adopt DEA efficient rule and to migrate that can induce the emergence of cooperation as the payoff is lower than its aspiration.

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