Enumerative and Distributional Results for $d$-combining Tree-Child Networks
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arXiv
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| Format: | Preprint |
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2022
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| author | Chang, Yu-Sheng Fuchs, Michael Liu, Hexuan Wallner, Michael Yu, Guan-Ru |
| author_facet | Chang, Yu-Sheng Fuchs, Michael Liu, Hexuan Wallner, Michael Yu, Guan-Ru |
| contents | Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks in which every reticulation node has exactly two parents. We extend these studies to $d$-combining tree-child networks where every reticulation node has now $d\geq 2$ parents, and we study one-component as well as general tree-child networks. For the number of one-component networks, we derive an exact formula from which asymptotic results follow that contain a stretched exponential for $d=2$, yet not for $d \geq 3$. For general networks, we find a novel encoding by words which leads to a recurrence for their numbers. From this recurrence, we derive asymptotic results which show the appearance of a stretched exponential for all $d \geq 2$. Moreover, we also give results on the distribution of shape parameters (e.g., number of reticulation nodes, Sackin index) of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves. We show phase transitions depending on $d$, leading to normal, Bessel, Poisson, and degenerate distributions. Some of our results are new even in the bicombining case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_03850 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Enumerative and Distributional Results for $d$-combining Tree-Child Networks Chang, Yu-Sheng Fuchs, Michael Liu, Hexuan Wallner, Michael Yu, Guan-Ru Combinatorics 05C20, 60C05, 60F05, 92D15 Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks in which every reticulation node has exactly two parents. We extend these studies to $d$-combining tree-child networks where every reticulation node has now $d\geq 2$ parents, and we study one-component as well as general tree-child networks. For the number of one-component networks, we derive an exact formula from which asymptotic results follow that contain a stretched exponential for $d=2$, yet not for $d \geq 3$. For general networks, we find a novel encoding by words which leads to a recurrence for their numbers. From this recurrence, we derive asymptotic results which show the appearance of a stretched exponential for all $d \geq 2$. Moreover, we also give results on the distribution of shape parameters (e.g., number of reticulation nodes, Sackin index) of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves. We show phase transitions depending on $d$, leading to normal, Bessel, Poisson, and degenerate distributions. Some of our results are new even in the bicombining case. |
| title | Enumerative and Distributional Results for $d$-combining Tree-Child Networks |
| topic | Combinatorics 05C20, 60C05, 60F05, 92D15 |
| url | https://arxiv.org/abs/2209.03850 |