Anomalous scaling in an age-dependent branching model

Keller-Schmidt, Stephanie; Tuğrul, Murat; Eguı́luz, Vı́ctor M.; Hernández-Garcı́a, Emilio; Klemm, Konstantin

doi:10.1103/physreve.91.022803

Cited by 7 publications

(5 citation statements)

References 43 publications

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“…All these balance indices depend only on the topology of the trees, not on the branch lengths or the actual labels on their leaves, although the balance of time-stamped trees has also been considered by Dearlove and Frost [11]. This abundance of balance indices is partly motivated by the advice given by Shao and Sokal [26] to use more than one such index to quantify the balance of a tree, as well as by their use as tools to test stochastic models of evolution [4,15,16,22,26]; other properties of the shapes of phylogenetic trees used in this connection include the distribution of clades' sizes [30,31] and the joint distribution of the numbers of rooted subtrees of different types [28].…”

Section: Introductionmentioning

confidence: 99%

A balance index for phylogenetic trees based on rooted quartets

et al. 2019

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We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's α-model and Aldous' β-model and on arbitrary trees under the α-γ-model. of isomorphisms of the restriction of T to them (the rooted quartet they define), and then we add up these values over all 4-tuples of different leaves of T . The idea behind the definition of this balance index is that a highly symmetrical evolutive process should give rise to symmetrical evolutive histories of many small subsets of taxa. In terms of phylogenetic trees, this leads us to expect that, the most symmetrical a phylogenetic tree is, the most symmetrical will be its restrictions to subsets of leaves of a fixed cardinality. Since the smallest number of leaves yielding enough different tree topologies to allow a meaningful comparison of their symmetry is 4, we assess the balance of a tree by measuring the symmetry of all its rooted quartets and adding up the results. And indeed, in Section 4 below we shall find the trees with maximum and minimum values of our rooted quartet index in both the arbitrary and the bifurcating cases, and it will turn out that the minimum value is reached exactly at the combs (see Fig. 1.(a)), which are usually considered the least balanced trees, and the maximum value is reached, in the arbitrary case, exactly at the rooted stars (see Fig. 1.(b)) and, in the bifurcating case, exactly at the maximally balanced trees (cf. Fig. 3 ), which in both cases are considered the most balanced trees.Besides taking its maximum and minimum values at the expected trees, other important features of our index are that it can be easily computed in linear time and that its mean value and variance can be explicitly computed on any probabilistic model of phylogenetic trees satisfying two natural conditions: independence under relabelings and sampling consistency. This allows us to provide these values for two well-known probabilistic models of bifurcating phylogenetic trees, Ford's α-model [13] and Aldous' β-model [2], which include as specific instances the Yule [14,29] and the uniform [6,24,19] models, as well as for Chen-Ford-Winkel's α-γ-model of multifurcating trees [7]. To our knowledge, this is the first shape index for which closed formulas for the expected value and the variance under the α-γ-model have been provided.The rest of this paper is organized as follows. In the next section we introduce the basic notations and facts on phylogenetic trees that will be used in the rest of the paper, and we recall several preliminary results on probabilistic models of phylogenetic trees, proving those results for which we have not been able to find a suitable reference in the literature. Then, in Section 3, we ...

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Section: Introductionmentioning

confidence: 99%

A balance index for phylogenetic trees based on rooted quartets

et al. 2019

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“…33], and Shao and Sokal [24, p. 1990] explicitly advise to use more than one such index to quantify tree balance. Such balance indices only depend on the topology of the trees, not on the branch lengths or the actual taxa labeling their leaves, and they have been widely used as tools to test stochastic models of evolution [3,12,13,18,24].…”

Section: Introductionmentioning

confidence: 99%

A balance index for phylogenetic trees based on quartets

Coronado

Mir

Rosselló

et al. 2018

Preprint

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We define a new balance index for phylogenetic trees based on the symmetry of the evolutive history of every quartet of leaves. This index makes sense for polytomic trees and it can be computed in time linear in the number of leaves. We compute its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance for bifurcating trees under Ford's α-model and Aldous' β-model and for arbitrary trees under the α-γ-model.

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“…In the emergence of cooperation, the persistence time in the learning of strategies in the spatial prisoner’s dilemma enhances cooperation and leads to heterogeneous distributions of persistence times 39 , and generates cyclic dominance of strategies 40 . In an evolutionary context, aging at the speciation events has been proposed as a mechanism to explain the shape of evolutionary trees 41 . Another example of the impact of aging can be found in the citation networks, where the age of the nodes has a crucial effect in the dynamic of growing of the network 42 .…”

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Competition in the presence of aging: dominance, coexistence, and alternation between states

Pérez

Klemm

Eguı́luz

2016

Sci Rep

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We study the stochastic dynamics of coupled states with transition probabilities depending on local persistence, this is, the time since a state has changed. When the system has a preference to adopt older states the system orders quickly due to the dominance of old states. When preference for new states prevails, the system can show coexistence of states or synchronized collective behavior resulting in long ordering times. In this case, the magnetization of the system oscillates around zero. Finally we discuss a potential application in social systems.

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Anomalous scaling in an age-dependent branching model

Cited by 7 publications

References 43 publications

A balance index for phylogenetic trees based on rooted quartets

A balance index for phylogenetic trees based on rooted quartets

A balance index for phylogenetic trees based on quartets

Competition in the presence of aging: dominance, coexistence, and alternation between states

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