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Theses

Random graphs in evolution

Abstract : This thesis consists of five independent research projects, related either to random graphs or to evolutionary biology - and most often to both. Chapters 2 and 3 introduce two random graphs that are obtained as the stationary distributions of graph-valued Markov chains. The first of these, which we term the split-and-drift random graph, aims to describe the structure and dynamics of interbreeding-potential networks; the second one is a random forest that was inspired by the celebrated Moran model of population genetics. Chapter 4 is devoted to the study of a new class of phylogenetic networks that we term ranked tree-child networks, or RTCNs for short. These networks correspond to a subclass of tree-child networks that are endowed with an additional structure ensuring compatibility with a time-embedded evolutionary process, and are designed to model reticulated phylogenies. We focus on the enumeration and sampling of RTCNs before turning to the structural properties of large uniform RTCNs. In Chapter 5, we prove a general result about oriented percolation in randomly oriented graphs: the positive association of the percolation cluster. Finally, in Chapter 6 we focus on a widely used statistic of populations: the mean age at which parents give birth. We point out several problems with one of the most widely used way to compute it, and provide an alternative measure.
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François Bienvenu. Random graphs in evolution. Combinatorics [math.CO]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS180⟩. ⟨tel-02932179⟩

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