Fixation probability on clique-based graphs

Choi, Jeong-Ok; Yu, Unjong

doi:10.1016/j.physa.2017.11.131

Cited by 7 publications

(9 citation statements)

References 12 publications

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“…In summary, there are reasons to accept the existence of a suppression mechanism shared by ℓ -graphs and clique-wheels (in the sense of [ 12 ] and [ 14 ]), although we think that new techniques of potential theory on directed graphs will be probably required to identify any mathematical underlying principle. For this purpose, ℓ -graphs have some interest since their state spaces (described explicitly in [ 7 ]) are simpler than those of the family of clique-wheels.…”

Section: Resultsmentioning

confidence: 99%

“…As before ℓ -graphs, friendship cycles and friendship stars seem to illustrate new mechanisms of suppression, whose underlying principles need to be studied and compared with those of clique-arms described in [ 14 ].…”

Section: Resultsmentioning

confidence: 99%

“…In fact, a family of graphs of order ≥ 12 that change from amplifier into suppressor as r increases has recently been shown in [ 14 ] by using numerical simulation. But, for the three graphs of order 7 that exhibit this change of regime, we know that there is a unique transition as we have symbolically computed their fixation probability for any r > 1.…”

Section: Introductionmentioning

confidence: 99%

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Evolutionary regime transitions in structured populations

2018

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The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. Usually, the population adopts the form of an undirected graph where the place occupied by each individual is represented by a vertex and it is bidirectionally linked to the places that can be occupied by its offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any vertex. But there also are undirected graph structures acting as suppressors of selection where this probability is less than that of the same individual placed in a homogeneous population. Here, firstly, we present the distribution of these evolutionary regimes for all undirected graphs with N ≤ 10 vertices. Some of them exhibit transitions between different regimes when the mutant fitness increases. In particular, as it has been already observed for small-order random graphs, we show that most graphs of order N ≤ 10 are amplifiers of selection. Secondly, we describe examples of amplifiers of order 7 that become suppressors from some critical value. In fact, for graphs of order N ≤ 7, we apply computer-aided techniques to symbolically compute their fixation probability and then their evolutionary regime, as well as the critical values for which they change their regime. Thirdly, the same technique is applied to some families of highly symmetrical graphs as a mean to explore methods of suppressing selection. The existence of suppression mechanisms that reverse an amplification regime when fitness increases could have a great interest in biology and network science. Finally, the analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, with multiple transitions between different regimes, which have not been examined in detail until now.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Evolutionary regime transitions in structured populations

2018

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“…As we already said, although transitions between different evolutionary regimes were known from [6], [9], [11] and [13], theses results reveal that undirected graphs have a complex and rich evolutionary dynamics, and pose new challenges in computing fixation probabilities and times. Now we know that numerical simulation cannot provide accurate answers to extremal problems on fixation probabilities, nor probably on absorption or fixation times.…”

Section: Discussionmentioning

confidence: 99%

“…Our results also demonstrate that numerical simulation is not enough to determine the evolutionary regime of a graph for the whole range of fitness values. Moreover, the very slight difference between different regimes for some graphs often forces to consider an extremely high number of trials when using Monte Carlo method, like recently in [13] to show a family of graphs of order ≥ 12 that changes from amplifier to suppressor as r increases. For example, 10 10 trials were needed in [7] to determine the evolutionary regime of the graphs N for 12 ≤ N ≤ 24 and 1 ≤ r ≤ 4.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in evolutionary dynamics

Ál

2018

Preprint

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The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. The population adopts the form of an undirected graph where the place occupied by each individual is represented by a node and it is bidirectionally linked to the places that can be occupied by its clonal offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any node. But there also are undirected graph structures acting as suppressors of selection where, on the contrary, the fixation probability of an advantageous mutant is less than that of the same individual placed in a homogeneous population. Here, we show that some undirected graphs exhibit phase transitions between both evolutionary regimes when the mutant fitness varies. Firstly, as was already observed for small order random graphs, we show that most graphs of order 10 or less are amplifiers of selection or suppressors that become amplifiers from a unique transition phase. Secondly, we give examples of amplifiers of order 7 that become suppressors from some critical value. For graphs of order 6 and 7, we apply computer aided techniques to exactly determine their fixation probabilities and then their evolutionary regimes, as well as the exact critical values for which each graph changes its regime. A similar technique is used to explore a general method to suppress selection in bigger orders, namely from 8 to 15, up to some critical fitness value. The analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, which have not been examined in detail until now, and poses some new challenges in computing fixation probabilities and times of evolutionary graphs.

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Effects of quadrilateral clustering on complex contagion

Jeong

Yu²

2022

Chaos, Solitons & Fractals

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Fixation probability on clique-based graphs

Cited by 7 publications

References 12 publications

Evolutionary regime transitions in structured populations

Evolutionary regime transitions in structured populations

Phase transitions in evolutionary dynamics

Effects of quadrilateral clustering on complex contagion

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