Search

In recent years, there has been an effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called $B_2$ index. In this paper, we study the $B_2$ index for a prominent class of phylogenetic networks: galled trees. We show that the $B_2$ index of a uniform leaf-labeled galled tree converges in distribution as the network becomes large. We characterize the corresponding limiting distribution, and show that its expected value is 2.707911858984... This is the first time that a balance index has been studied to this level of detail for a random phylogenetic network. One specificity of this work is that we use two different and independent approaches, each with its advantages: analytic combinatorics, and local limits. The analytic combinatorics approach is more direct, as it relies on standard tools; but it involves slightly more complex calculations. Because it has not previously been used to study such questions, the local limit approach requires developing an extensive framework beforehand; however, this framework is interesting in itself and can be used to tackle other similar problems.

The Sackin index is an important measure for the balance of phylogenetic trees. We investigate two extensions of the Sackin index to the class of galled trees and two of its subclasses (simplex galled trees and normal galled trees) where we consider both labeled and unlabeled galled trees. In all cases, we show that the mean of the Sackin index for a network which is uniformly sampled from its class is asymptotic to $\mu n^{3/2}$ for an explicit constant $\mu$. In addition, we show that the scaled Sackin index convergences weakly and with all its moments to the Airy distribution., Comment: Dedicated to Michael Drmota on the occasion of his 60th birthday. 25 pages

This paper addresses the significant challenge of recognizing behaviors in non-human primates, specifically focusing on chimpanzees. Automated behavior recognition is crucial for both conservation efforts and the advancement of behavioral research. However, it is significantly hindered by the labor-intensive process of manual video annotation. Despite the availability of large-scale animal behavior datasets, the effective application of machine learning models across varied environmental settings poses a critical challenge, primarily due to the variability in data collection contexts and the specificity of annotations. In this paper, we introduce ChimpBehave, a novel dataset featuring over 2 hours of video (approximately 193,000 video frames) of zoo-housed chimpanzees, meticulously annotated with bounding boxes and behavior labels for action recognition. ChimpBehave uniquely aligns its behavior classes with existing datasets, allowing for the study of domain adaptation and cross-dataset generalization methods between different visual settings. Furthermore, we benchmark our dataset using a state-of-the-art CNN-based action recognition model, providing the first baseline results for both within and cross-dataset settings. The dataset, models, and code can be accessed at: https://github.com/MitchFuchs/ChimpBehave, Comment: CV4Animals: Computer Vision for Animal Behavior Tracking and Modeling In conjunction with Computer Vision and Pattern Recognition 2024

We propose the class of galled tree-child networks which is obtained as intersection of the classes of galled networks and tree-child networks. For the latter two classes, (asymptotic) counting results and stochastic results have been proved with very different methods. We show that a counting result for the class of galled tree-child networks follows with similar tools as used for galled networks, however, the result has a similar pattern as the one for tree-child networks. In addition, we also consider the (suitably scaled) numbers of reticulation nodes of random galled tree-child networks and show that they are asymptotically normal distributed. This is in contrast to the limit laws of the corresponding quantities for galled networks and tree-child networks which have been both shown to be discrete., Comment: Submitted to AofA2024

We give exact and asymptotic counting results for the number of galled networks and reticulation-visible networks with few reticulation vertices. Our results are obtained with the component graph method, which was introduced by L. Zhang and his coauthors, and generating function techniques. For galled networks, we in addition use analytic combinatorics. Moreover, in an appendix, we consider maximally reticulated reticulation-visible networks and derive their number, too., Comment: Revised version; accepted for publication in Bull. Math. Biol

Given a gene-tree labeled topology $G$ and a species tree $S$, the "ancestral configurations" at an internal node $k$ of $S$ represent the combinatorially different sets of gene lineages that can be present at $k$ when all possible realizations of $G$ in $S$ are considered. Ancestral configurations have been introduced as a data structure for evaluating the conditional probability of a gene-tree labeled topology given a species tree, and their enumeration assists in describing the complexity of this computation. In the case that the gene-tree labeled topology $G=t$ matches that of the species tree $S$, by techniques of analytic combinatorics, we study distributional properties of the "total" number of ancestral configurations measured across the different nodes of a random labeled topology $t$ selected under the uniform and the Yule probability models. Under both of these probabilistic scenarios, we show that the total number $T_n$ of ancestral configurations of a random labeled topology of $n$ taxa asymptotically follows a lognormal distribution. Over uniformly distributed labeled topologies, the asymptotic growth of the mean and the variance of $T_n$ are found to satisfy $\mathbb{E}_{\rm U}[T_n] \sim 2.449 \cdot 1.333^n$ and $\mathbb{V}_{\rm U}[T_n] \sim 5.050 \cdot 1.822^n$, respectively. Under the Yule model, which assigns higher probabilities to more balanced labeled topologies, we obtain the mean $\mathbb{E}_{\rm Y}[T_n] \sim 1.425^n$ and the variance $\mathbb{V}_{\rm Y}[T_n] \sim 2.045^n$.

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., Comment: Revised version; accepted for publication in Adv. in Appl. Math

The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -- also called "histories" -- with a finite number $n$ of leaves. We study the lengths $\ell_1 > \ell_2 > ... > \ell_k > ...$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n \rightarrow \infty$, we show that the random variable $\ell_k$, once rescaled as $\frac{n-\ell_k}{\sqrt{n/2}}$, follows a $\chi$-distribution with $2k$ degrees of freedom, with mean $\mathbb E(\ell_k) \sim n$ and variance $\mathbb V(\ell_k) \sim n \big(k-\frac{\pi k^2}{16^k} \binom{2k}{k}^2\big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.

We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height $1$ and $2$, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to $0$ and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern., Comment: 20 pages

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 {\it bicombining tree-child networks} which are tree-child networks with every reticulation node having exactly two parents. In this paper, we extend these studies to {\it $d$-combining tree-child networks} where every reticulation node has now $d\geq 2$ parents. Moreover, we also give results and conjectures on the distributional behavior of the number of reticulation nodes of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves., Comment: Extended abstract which is accepted for presentation at AofA2022; an accompanying Maple worksheet can be found in the ancillary file folder

The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of a tree shape (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short elementary proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted trees with leaves being labelled with taxa) under the uniform model., Comment: 10 pages, 1 fugre

Tree-child networks are an important network class which are used in phylogenetics to model reticulate evolution. In a recent paper, Pons and Batle (2021) conjectured a relation between tree-child networks and certain words. In this short note, we prove their conjecture for the (important) class of one-component tree-child networks., Comment: 6 pages

The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper., Comment: revised version, 12 pages, 13 Figures

Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices has the first-order asymptotics \[ {\rm TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad (\ell\rightarrow\infty). \] Moreover, they also computed $c_k$ for $k=1,2,$ and $3$. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for $c_k$, namely, $c_k=2^{k-1}\sqrt{2}/k!$ for all $k\geq 0$., Comment: accepted version, 13 pages

Graph tries are a new and interesting data structure proposed by Jacquet in 2014. They generalize the classical trie data structure which has found many applications in computer science and is one of the most popular data structure on words. For his generalization, Jacquet considered the size (or space requirement) and derived an asymptotic expansion for the mean and the variance when graph tries are built from $n$ independently chosen random labelings of a rooted $M$-ary tree. Moreover, he conjectured a central limit theorem for the (suitably normalized) size as the number of labelings tends to infinity. In this paper, we verify this conjecture with the method of moments., Comment: 20 pages; some minor errors fixed

We show a first-order asymptotics result for the number of galled networks with $n$ leaves. This is the first class of phylogenetic networks of {\it large} size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of a galled networks with $n$ leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) which was devised for the purpose of (exactly) counting galled networks. Moreover, an old result of Bender and Richmond (1984) plays a crucial role in our proofs, too., Comment: 23 pages; revised version

In previous work, we gave asymptotic counting results for the number of tree-child and normal networks with $k$ reticulation vertices and explicit exponential generating functions of the counting sequences for $k=1,2,3$. The purpose of this note is two-fold. First, we make some corrections to our previous approach which overcounted the above numbers and thus gives erroneous exponential generating functions (however, the overcounting does not effect our asymptotic counting results). Secondly, we use our (corrected) exponential generating functions to derive explicit formulas for the number of tree-child and normal networks with $k=1,2,3$ reticulation vertices. This re-derives recent results of Carona and Zhang, answers their question for normal networks with $k=2$, and adds new formulas in the case $k=3$., Comment: revised version; 20 pages

For a pair consisting of a gene tree and a species tree, the ancestral configurations at an internal node of the species tree are the distinct sets of gene lineages that can be present at that node. Ancestral configurations appear in computations of gene tree probabilities under evolutionary models conditional on fixed species trees, and the enumeration of root ancestral configurations -- ancestral configurations at the root of the species tree -- assists in describing the complexity of these computations. In the case that the gene tree matches the species tree in topology, we study the distribution of the number of root ancestral configurations of a random labeled tree topology under each of two models., Comment: 28 pages, 9 figures

In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with $n$ leaves has the factor $n^{2n}$ in its main asymptotic growth term. In this paper, we improve this by completely identifying the main asymptotic growth term up to a constant. More precisely, we show that the number of tree-child networks with $n$ leaves grows like \[ \Theta\left(n^{-2/3}e^{a_1(3n)^{1/3}}\left(\frac{12}{e^2}\right)^{n}n^{2n}\right), \] where $a_1=-2.338107410\cdots$ is the largest root of the Airy function of first kind. For the proof, we bijectively map the underlying graph-theoretical problem onto a problem on words. For the latter, we can find a recurrence to which a recent powerful asymptotic method of Elvey Price, Fang, and Wallner (2019) can be applied., Comment: 20 pages; this is the revised version incorporating the comments of the referees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.

For SCFTs with an $SU(2)$ R-symmetry, we determine the superconformal blocks that contribute to the four-point correlation function of a priori distinct half-BPS superconformal primaries as an expansion in terms of the relevant bosonic conformal blocks. This is achieved by using the superconformal Casimir equation and the superconformal Ward identity to fix the coefficients of the bosonic blocks uniquely in a dimension-independent way. In addition we find that many of the resulting coefficients are related through a web of linear transformations of the conformal data., Comment: 42 pages, ancillary Mathematica notebook listing all block coefficients; minor corrections, published version

In recent decades, phylogenetic networks have become a standard tool in modeling evolutionary processes. Nevertheless, basic combinatorial questions about them are still largely open. For instance, even the asymptotic counting problem for the class of phylogenetic networks and subclasses is unsolved. In this paper, we propose a method based on generating functions to count networks with few reticulation vertices for two subclasses which are important in applications: tree-child networks and normal networks. In particular, our method can be used to completely solve the asymptotic counting problem for these network classes when the number of reticulation vertices remains fixed and the network size tends to infinity., Comment: second revision of our previous submission; 28 pages

We give a detailed asymptotic analysis of the profiles of random symmetric digital search trees, which are in close connection with the performance of the search complexity of random queries in such trees. While the expected profiles have been analyzed for several decades, the analysis of the variance turns out to be very difficult and challenging, and requires the combination of several different analytic techniques, including Mellin and Laplace transforms, analytic de-Poissonization, and Laplace convolutions. Our results imply concentration of the profiles in the range where the mean tends to infinity. Moreover, we also obtain a two-point concentration for the distributions of the height and the saturation level., Comment: The central limit theorem was removed from this version (and moved to a follow-up paper) since the proof in the previous versions was incomplete. Also, the word "Concentration Properties" was added to the title since this part now entirely focuses on such results

It is believed that any classical gauge symmetry gives rise to an L$_\infty$ algebra. Based on the recently realized relation between classical ${\cal W}$ algebras and L$_\infty$ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum ${\cal W}$ algebras, we provide a physically well motivated definition of quantum L$_\infty$ algebras describing the consistency of global symmetries in quantum field theories. In this case we are restricted to only two non-trivial graded vector spaces $X_0$ and $X_{-1}$ containing the symmetry variations and the symmetry generators. This quantum L$_\infty$ algebra structure is explicitly exemplified for the quantum ${\cal W}_3$ algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L$_\infty$ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L$_\infty$ algebra of closed string field theory., Comment: 21 pages, v2: clarifications, version published in JHEP

It is shown that the closure of the infinitesimal symmetry transformations underlying classical ${\cal W}$ algebras give rise to L$_\infty$ algebras with in general field dependent gauge parameters. Therefore, the class of well understood ${\cal W}$ algebras provides highly non-trivial examples of such strong homotopy Lie-algebras. We develop the general formalism for this correspondence and apply it explicitly to the classical ${\cal W}_3$ algebra., Comment: 15 pages; v2: typos corrected, minor changes

Recent progress on the relation between asymmetric conformal field theories and vacua of gauged supergravities is reviewed. This includes an attempt to classify asymmetric Gepner models in 8D, 6D and 4D with at least eight supercharges, and how they can be categorized by just a few mechanisms including the super Higgs-effect. The latter is a prerequisite for the identification with non-geometric flux compactifications. We also highlight our findings for the identification of 4D asymmetric Gepner models with ${\cal N}=1$ supersymmetry with Minkowski vacua of ${\cal N}=2$ GSUGRA., Comment: Proceedings prepared for the "Workshop on Geometry and Physics", November 2016, Ringberg Castle, Germany; v2: references added

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose correlation is $0$, $1$ and periodically oscillating. Moreover, the same type of behaviors is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries., Comment: 21 pages; journal version of arXiv:1604.08658

We study asymmetric simple-current extensions of Gepner models in dimensions $D=4,6,8$ with at least eight supercharges in the right-moving sector. The models obtained in an extensive stochastic computer search belong to a small number of different classes. These classes can be categorized as dimensional reductions, asymmetric orbifolds with $(-1)^{F_L}$, extra gauge enhancement and as coming from the super Higgs-effect. Models in the latter class are particularly interesting, as they may correspond to non-geometric flux compactifications., Comment: 37 pages