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

2021 ◽  
Vol 82 (7) ◽  
Author(s):  
Hendrik Richter
Keyword(s):  
Spectral Analysis ◽  
Structural Properties ◽  
Evolutionary Dynamics ◽  
Mixed Population ◽  
Fixation Probability ◽  
Regular Graphs ◽  
Central Question ◽  
Random Walks On Graphs ◽  
Determining Factor ◽  
Coalescence Times

AbstractA central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

SPECTRAL ANALYSIS FOR WEIGHTED ITERATED TRIANGULATIONS OF GRAPHS

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850017 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Spectral Analysis ◽  
Closed Form ◽  
Structural Properties ◽  
Spanning Trees ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Successive Generations

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

scholarly journals How life history can sway the fixation probability of mutants

2016 ◽  
Author(s):  
Xiang-Yi Li ◽  
Shun Kurokawa ◽  
Stefano Giaimo ◽  
Arne Traulsen
Keyword(s):  
Life History ◽  
Age Structure ◽  
Evolutionary Dynamics ◽  
Selective Advantage ◽  
Fixation Probability ◽  
Relative Importance ◽  
Demographic Structure ◽  
Frequency Dependent ◽  
Intermediate Fraction ◽  
Size And Structure

AbstractIn this work, we study the effects of demographic structure on evolutionary dynamics, when selection acts on reproduction, survival, or both. In contrast with the previously discovered pattern that the fixation probability of a neutral mutant decreases while population becomes younger, we show that a mutant with constant selective advantage may have a maximum or a minimum of the fixation probability in populations with an intermediate fraction of young individuals. This highlights the importance of life history and demographic structure in studying evolutionary dynamics. We also illustrate the fundamental differences between selection on reproduction and on survival when age structure is present. In addition, we evaluate the relative importance of size and structure of the population in determining the fixation probability of the mutant. Our work lays the foundation for studying also density and frequency dependent effects in populations when demographic structures cannot be neglected.

scholarly journals The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

2021 ◽  
Vol 17 (10) ◽  
pp. e1009537
Author(s):  
Mohammad Ali Dehghani ◽  
Amir Hossein Darooneh ◽  
Mohammad Kohandel
Keyword(s):  
Network Structure ◽  
Evolutionary Dynamics ◽  
Finite Population ◽  
Small World ◽  
Fixation Time ◽  
Fixation Probability ◽  
Neutral Drift ◽  
Scale Free ◽  
Wide Range ◽  
And Mathematics

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

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 On the fixation probability of superstars

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130193 ◽  
Author(s):  
Josep Díaz ◽  
Leslie Ann Goldberg ◽  
George B. Mertzios ◽  
David Richerby ◽  
Maria Serna ◽  
...  
Keyword(s):  
Computer Algebra ◽  
Evolutionary Dynamics ◽  
Process Models ◽  
Relative Fitness ◽  
Genetic Mutations ◽  
Fixation Probability ◽  
Mutant Population ◽  
Moran Process ◽  
Limiting Behaviour ◽  
Symbolic Matrix

The Moran process models the spread of genetic mutations through populations. A mutant with relative fitness r is introduced and the system evolves, either reaching fixation (an all-mutant population) or extinction (no mutants). In a widely cited paper, Lieberman et al. (2005 Evolutionary dynamics on graphs. Nature 433 , 312–316) generalize the model to populations on the vertices of graphs. They describe a class of graphs (‘superstars’), with a parameter k and state that the fixation probability tends to 1− r − k as the graphs get larger: we show that this is untrue as stated. Specifically, for k =5, we show that the fixation probability (in the limit, as graphs get larger) cannot exceed 1−1/ j ( r ), where j ( r )= Θ ( r 4 ), contrary to the claimed result. Our proof is fully rigorous, though we use a computer algebra package to invert a 31×31 symbolic matrix. We do believe the qualitative claim of Lieberman et al. —that superstar fixation probability tends to 1 as k increases—and that it can probably be proved similarly to their sketch. We were able to run larger simulations than the ones they presented. Simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour.

Punctuated Change

2021 ◽  
pp. 95-110
Author(s):  
Daniel A. Levinthal
Keyword(s):  
Technological Change ◽  
Evolutionary Dynamics ◽  
Critical Role ◽  
Rapid Change ◽  
Organizational Capabilities ◽  
Organizational Strategy ◽  
Change Processes ◽  
Central Question ◽  
Multi Level

The pace of change is a central question regarding evolutionary dynamics. Some management theorists have pointed to processes of punctuated change; however, it is argued here that such accounts have generally under-attended to the multi-level nature of these processes and in particular to the critical role of speciation. By recognizing the multi-level nature of these dynamics, we can reconcile our often conflicting sense of organizations and technologies as undergoing periods of rapid change, while still conforming to a gradualist perspective with regard to the underlying elements of organizational capabilities and technologies. This argument is developed to consider change processes in three different contexts: the pace of technological change, shifts in organizational strategy and capabilities, and changes in the scope of firms.

Spectral analysis for weighted iterated q-triangulations of graphs

2020 ◽  
Vol 31 (03) ◽  
pp. 2050042
Author(s):  
Yufei Chen ◽  
Wenxia Li
Keyword(s):  
Spectral Analysis ◽  
Theoretical Analysis ◽  
Closed Form ◽  
Structural Properties ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Successive Generations

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighed iterated [Formula: see text]-triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As examples of application of these results, we then derive closed-form expressions for their Kemeny’s constant and multiplicative Kirchhoff index. Simulation example is also provided to demonstrate the effectiveness of the theoretical analysis.

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