The analysis of bi-level evolutionary graphs

Zhang, Pei-ai; Nie, Pu‐yan; Hu, D.-Q.; Zou, Feiyan

doi:10.1016/j.biosystems.2007.05.008

Cited by 13 publications

(12 citation statements)

References 11 publications

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“…Many researchers have investigated individuals that play classic evolutionary games while being constrained by the topology of a graph [5,[17][18][19][20][21]. Many evolutionary biological processes can be modelled as bi-level graphs, including examples of symbiosis and commensalism [22]. Outside of evolutionary biology, graph theory or similar concepts have been applied to model economic networks [23], the spread of epidemics in populations [24] and brain connectivity networks [25].…”

Section: Discussionmentioning

confidence: 99%

Martingales and fixation probabilities of evolutionary graphs

2014

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Evolutionary graph theory is the study of birthdeath processes that are constrained by population structure. A principal problem in evolutionary graph theory is to obtain the probability that some initial population of mutants will fixate on a graph, and to determine how that fixation probability depends on the structure of that graph. A fluctuating mutant population on a graph can be considered as a random walk. Martingales exploit symmetry in the steps of a random walk to yield exact analytical expressions for fixation probabilities. They do not require simplifying assumptions such as large population sizes or weak selection. In this paper, we show how martingales can be used to obtain fixation probabilities for symmetric evolutionary graphs. We obtain simpler expressions for the fixation probabilities of star graphs and complete bipartite graphs than have been previously reported and show that these graphs do not amplify selection for advantageous mutations under all conditions.

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Section: Discussionmentioning

confidence: 99%

Martingales and fixation probabilities of evolutionary graphs

2014

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“…The evolutionary graph was initially introduced in 2005 by Lieberman et al [1] and significantly developed by Nie and Zhang [3][4][5], Zhang et al [9,10], Ohtsuki et al [2], and Shakarian et al [6] and the references therein. In this framework, the structure of a population is modeled by a weighted digraph on vertices 1, 2, .…”

Section: The Fixation Probabilities Of Single-level Egsmentioning

confidence: 99%

“…Nie explained the phenomenon of autoeciousness. Zhang et al [9] discussed a class of bilevel EGs: the upper levels are isothermal and the lower levels are one rooted. The bilevel EGs reflect hierarchical population structures.…”

Section: The Fixation Probabilities Of a Class Of Bilevel Egsmentioning

confidence: 99%

“…Some EGs are sensitive to the positions where the new mutants appear. Some papers as Lieberman et al [1], Shakarian et al [6], Nowak et al (2006), Nie [3], and Zhang et al [9,10] have noticed that the positions of the new appeared mutants in EGs are important to their fixation probabilities. For the directed line, only the mutant appearing at the origin vertex can take over the whole population.…”

Section: Introductionmentioning

confidence: 99%

“…The fixation probabilities of some multilevel EGs depend on the appearance of the levels of the new mutants. In this paper, we lay emphasis on the position of the new arisen mutant and put forward a way to calculate the fixation probability of an EG and calculate them for the directed lines, directed circles, and a class of bilevel EGs which is discussed by Zhang et al [9] and correct their results.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

2014

Journal of Applied Mathematics

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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.

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