arXiv subjects

Noah A. Rosenberg

Publications and source records attributed to Noah A. Rosenberg.

At least 19 recordsLinked to original sources

arXiv2026

Quo nomine vis vocari? Random-copying, conformity, and anti-conformity in the temporal sequence of papal names

The study of cultural evolution seeks to understand the processes by which behavioral variants are chosen in human cultures over time. The selection of new popes, each of whom chooses a papal name -- typically reusing previous names in reference to previous popes -- is among the longest ongoing cultural processes taking place in a single human institution. Here, we test the papal naming sequence against predictions of models developed in population genetics and stochastic processes -- Ewens sampling theory and the Chinese restaurant process -- which in the case of papal names amount to randomly copying an existing name in proportion to its frequency, with the possibility of innovation. Although papal name choices are careful individual decisions, we find that the sequence of papal names often accords with a random-copying model, with long periods in which a model of conformity in name choices more closely fits the name sequence and other periods in which anti-conformity provides a closer fit. Hence, despite the consideration that enters into individual choices, aggregate cultural behavior in a 2000-year old human process can potentially be described with simple laws. Our methods can potentially be used to deepen quantitative analyses of conformity and anti-conformity in other cultural processes that involve discrete cultural variants.

q-bio.PE↗

arXiv2026

Extremal ranks of unlabeled multifurcating rooted trees in a bijective encoding by the positive integers

Maranca and Rosenberg (2024) devised a ranking scheme for unlabeled multifurcating rooted trees, in which the trees are bijectively associated with the positive integers. Here, generalizing earlier results for bifurcating trees, we determine, for trees with a fixed number of leaves, which multifurcating trees obtain the maximal and minimal ranks. We identify these maximizing and minimizing trees for each of two sets of unlabeled multifurcating rooted trees: strictly $k$-furcating trees, in which each internal node possesses exactly $k$ descendants, and at-most-$k$-furcating trees, in which internal nodes possess at least 2 and at most $k$ descendants. In both scenarios, we find that a tree that can be regarded as maximally balanced attains the minimal rank, and a minimally balanced tree attains the maximal rank. We deduce recurrences for the maximal and minimal rank for trees with fixed numbers of leaves in both the strictly $k$-furcating and at-most-$k$-furcating cases. The maximal rank on $(n-1)(k-1)+1$ leaves grows with $(k!)^{\frac{1}{k-1}} β_k^{(k^n)}$ in the strictly $k$-furcating case, and the maximal rank on $n$ leaves grows with $(k!)^{\frac{1}{k-1}} γ_k^{(k^n)}$ in the at-most-$k$-furcating case, where $β_k > 1$ and $γ_k > 1$ are constants that depend on the value of $k$. We show that $β_k$ decreases as the value of $k$ increases, and that $γ_k > β_k$ for $k \geq 3$. The results contribute to the use of tree encodings for empirical characterization of phylogenies and measurement of tree balance.

math.CO↗

arXiv2026

Maximally probable tree topologies with $r$-furcation

For a specific rooted labeled tree topology, a labeled history is a sequence of branchings that give rise to that labeled topology as it unfolds over time. Here, for $r$-furcating trees, we use a connection with Huffman trees from information theory to identify maximally probable rooted trees -- unlabeled $r$-furcating topologies whose labelings each have a number of labeled histories greater than or equal to those of all other labeled topologies. Our characterization of the unique maximally probable $r$-furcating unlabeled topology generalizes the Harding--Hammersley--Grimmett result identifying the maximally probable bifurcating unlabeled topology, and it provides a new proof for that result. We present a conjecture for the maximally probable $r$-furcating unlabeled topology if labeled histories are tabulated allowing for simultaneous branching events across multiple internal nodes of a tree.

math.CO↗

arXiv2026

Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees

Rooted binary phylogenetic networks are extensions of rooted binary trees, adding reticulation nodes that are designed to represent evolutionary processes that involve hybridization events. Enumerative combinatorics studies have counted leaf-labeled phylogenetic networks in a variety of classes, finding that when the number of reticulations is fixed, the time-consistent galled trees are asymptotically less numerous than each of several network classes that had been previously examined. Here we provide enumerative results on two additional network classes: general galled trees and simplex time-consistent galled trees. We show that for a fixed number of galls, as the number of leaves goes to infinity, the asymptotic count of general galled trees is identical to that of time-consistent galled trees, whereas the count of simplex time-consistent galled trees is smaller. If the number of galls is not restricted, then the asymptotic approximations all differ: simplex time-consistent galled trees are less numerous than time-consistent galled trees, which are in turn less numerous than general galled trees. We also report a variety of additional results: recursions to count the studied networks with small numbers of leaves a fixed number of galls, as well as enumerative results for unlabeled networks in the classes that we investigate.

math.CO↗

arXiv2025

Labeled histories and maximally probable labeled topologies with multifurcation

In mathematical phylogenetics, labeled histories describe the sequences by which sets of labeled lineages coalesce to a shared ancestral lineage. We study labeled histories for at-most-$r$-furcating trees. Consider a rooted leaf-labeled tree in which internal nodes each have $i$ offspring, and $i$ is permitted to range from 2 to $r$ across internal nodes, for a specified value of $r$. For labeled topologies with $n$ leaves, we enumerate the total number of labeled histories with at-most-$r$-furcation. We enumerate the labeled histories possessed by a specific at-most-$r$-furcating labeled topology. We then demonstrate that the maximally probable at-most-$r$-furcating unlabeled topology on $n \geq 2$ leaves -- the unlabeled topology whose labelings have the largest number of labeled histories -- is the maximally probable strictly bifurcating unlabeled topology on $n$ leaves. Finally, we enumerate labeled histories for at-most-$r$-furcating labeled topologies in a setting that permits simultaneous branchings. We similarly reduce the problem of identifying the maximally probable at-most-$r$-furcating unlabeled topology on $n \geq 2$ leaves, allowing simultaneity, to that of identifying the maximally probable strictly bifurcating unlabeled topology on $n$ leaves, with simultaneity; we conjecture the shape of this bifurcating unlabeled topology. The computations contribute to the study of multifurcation, which arises in various biological processes, and they connect to analogous mathematical settings involving precedence-constrained scheduling.

q-bio.PE↗

arXiv2025

An nth-cousin mating model and the n-anacci numbers

In seeking to understand the size of inbred pedigrees, J. Lachance (J. Theor. Biol. 261, 238-247, 2009) studied a population model in which, for a fixed value of $n$, each mating occurs between $n$th cousins. We explain a connection between the second-cousin case of the model ($n=2$) and the Fibonacci sequence, and more generally, between the $n$th-cousin case and the $n$-anacci sequence $(n \geq 2)$. For a model with $n$th-cousin mating $(n \geq 1)$, we obtain the generating function describing the size of the pedigree $t$ generations back from the present, and we use it to evaluate the asymptotic growth of the pedigree size. In particular, we show that the growth of the pedigree asymptotically follows the growth rate of the $n$-anacci sequence -- the golden ratio $ϕ= (1 + \sqrt{5})/2 \approx 1.6180$ in the second-cousin case $n=2$ -- and approaches 2 as $n$ increases. The computations explain the appearance of familiar numerical sequences and constants in a pedigree model. They also recall similar appearances of such sequences and constants in studies of population biology more generally.

q-bio.PE↗

arXiv2025

The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains

Coalescent models of bifurcating genealogies are used to infer evolutionary parameters from molecular data. However, there are many situations where bifurcating genealogies do not accurately reflect the true underlying ancestral history of samples, and a multifurcating genealogy is required. The space of multifurcating genealogical trees, where nodes can have more than two descendants, is largely underexplored in the setting of coalescent inference. In this paper, we examine the space of rooted, ranked, and unlabeled multifurcating trees. We recursively enumerate the space and then construct a partial ordering which induces a lattice on the space of multifurcating ranked tree shapes. The lattice structure lends itself naturally to defining Markov chains that permit exploration on the space of multifurcating ranked tree shapes. Finally, we prove theoretical bounds for the mixing time of two Markov chains defined on the lattice, and we present simulation results comparing the distribution of trees and tree statistics under various coalescent models to the uniform distribution on this tree space.

math.PR↗

arXiv2025

Enumerative combinatorics of unlabeled and labeled time-consistent galled trees

In mathematical phylogenetics, the time-consistent galled trees provide a simple class of rooted binary network structures that can be used to represent a variety of different biological phenomena. We study the enumerative combinatorics of unlabeled and labeled time-consistent galled trees. We present a new derivation via the symbolic method of the number of unlabeled time-consistent galled trees with a fixed number of leaves and a fixed number of galls. We also derive new generating functions and asymptotics for labeled time-consistent galled trees.

math.CO↗

arXiv2024

Enumeration of rooted binary perfect phylogenies

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We address a variety of enumerative problems concerning rooted binary perfect phylogenies with sample size $s$: the rooted binary unlabeled trees in which a sample of size $s$ lineages is distributed across the leaves of an unlabeled tree with $n$ leaves, $1 \leq n \leq s$. The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.

q-bio.PE↗

arXiv2024

Combinatorics of a dissimilarity measure for pairs of draws from discrete probability vectors on finite sets of objects

Motivated by a problem in population genetics, we examine the combinatorics of dissimilarity for pairs of random unordered draws of multiple objects, with replacement, from a collection of distinct objects. Consider two draws of size $K$ taken with replacement from a set of $I$ objects, where the two draws represent samples from potentially distinct probability distributions over the set of $I$ objects. We define the set of \emph{identity states} for pairs of draws via a series of actions by permutation groups, describing the enumeration of all such states for a given $K \geq 2$ and $I \geq 2$. Given two probability vectors for the $I$ objects, we compute the probability of each identity state. From the set of all such probabilities, we obtain the expectation for a dissimilarity measure, finding that it has a simple form that generalizes a result previously obtained for the case of $K=2$. We determine when the expected dissimilarity between two draws from the same probability distribution exceeds that of two draws taken from different probability distributions. We interpret the results in the setting of the genetics of polyploid organisms, those whose genetic material contains many copies of the genome ($K > 2$).

math.CO↗

arXiv2024

Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees

The Colijn--Plazzotta ranking is a bijective encoding of the unlabeled binary rooted trees with positive integers. We show that the rank $f(t)$ of a tree $t$ is closely related to its height $h$, the length of the longest path from a leaf to the root. We consider the rank $f(τ_n)$ of a random $n$-leaf tree $τ_n$ under each of three models: (i) uniformly random unlabeled unordered binary rooted trees, or unlabeled topologies; (ii) uniformly random leaf-labeled binary trees, or labeled topologies under the uniform model; and (iii) random binary search trees, or labeled topologies under the Yule--Harding model. Relying on the close relationship between tree rank and tree height, we obtain results concerning the asymptotic properties of $\log \log f(τ_n)$. In particular, we find $\mathbb{E} \{\log_2 \log f(τ_n)\} \sim 2 \sqrt{πn}$ for uniformly random unlabeled ordered binary rooted trees and uniformly random leaf-labeled binary trees, and for a constant $α\approx 4.31107$, $\mathbb{E}\{\log_2 \log f(τ_n)\} \sim α\log n $ for leaf-labeled binary trees under the Yule--Harding model. We show that the mean of $f(τ_n)$ itself under the three models is largely determined by the rank $c_{n-1}$ of the highest-ranked tree -- the caterpillar -- obtaining an asymptotic relationship with $π_n c_{n-1}$, where $π_n$ is a model-specific function of $n$. The results resolve open problems, providing a new class of results on an encoding useful in mathematical phylogenetics.

math.CO↗

arXiv2023

A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees

To a given gene tree topology $G$ and species tree topology $S$ with leaves labeled bijectively from a fixed set $X$, one can associate a set of ancestral configurations, each of which encodes a set of gene lineages that can be found at a given node of the species tree. We introduce a lattice structure on ancestral configurations, studying the directed graphs that provide graphical representations of lattices of ancestral configurations. For a matching gene tree topology and species tree topology $G=S$, we present a method for defining the digraph of ancestral configurations from the tree topology by using iterated cartesian products of graphs. We show that a specific set of paths on the digraph of ancestral configurations is in bijection with the set of labeled histories -- a well-known phylogenetic object that enumerates possible temporal orderings of the coalescences of a tree. For each of a series of tree families, we obtain closed-form expressions for the number of labeled histories by using this bijection to count paths on associated digraphs. Finally, we prove that our lattice construction extends to nonmatching tree pairs, and we use it to characterize pairs $(G,S)$ having the maximal number of ancestral configurations for a fixed $G$. We discuss how the construction provides new methods for performing enumerations of combinatorial aspects of gene and species trees.

q-bio.PE↗

arXiv2023

Cultural transmission of move choice in chess

The study of cultural evolution benefits from detailed analysis of cultural transmission in specific human domains. Chess provides a platform for understanding the transmission of knowledge due to its active community of players, precise behaviors, and long-term records of high-quality data. In this paper, we perform an analysis of chess in the context of cultural evolution, describing multiple cultural factors that affect move choice. We then build a population-level statistical model of move choice in chess, based on the Dirichlet-multinomial likelihood, to analyze cultural transmission over decades of recorded games played by leading players. For moves made in specific positions, we evaluate the relative effects of frequency-dependent bias, success bias, and prestige bias on the dynamics of move frequencies. We observe that negative frequency-dependent bias plays a role in the dynamics of certain moves, and that other moves are compatible with transmission under prestige bias or success bias. These apparent biases may reflect recent changes, namely the introduction of computer chess engines and online tournament broadcasts. Our analysis of chess provides insights into broader questions concerning how social learning biases affect cultural evolution.

physics.soc-ph↗

arXiv2023

The distributions under two species-tree models of the total number of ancestral configurations for matching gene trees and species trees

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 $\mathbb{V}_{\rm Y}[T_n] \sim 2.045^n$.

math.PR↗

arXiv2022

On the number of genealogical ancestors tracing to the source groups of an admixed population

In genetically admixed populations, admixed individuals possess ancestry from multiple source groups. Studies of human genetic admixture frequently estimate ancestry components corresponding to fractions of individual genomes that trace to specific ancestral populations. However, the same numerical ancestry fraction can represent a wide array of admixture scenarios. Using a mechanistic model of admixture, we characterize admixture genealogically: how many distinct ancestors from the source populations does the admixture represent? We consider African Americans, for whom continent-level estimates produce a 75-85% value for African ancestry on average and 15-25% for European ancestry. Genetic studies together with key features of African-American demographic history suggest ranges for model parameters. Using the model, we infer that if genealogical lineages of a random African American born during 1960-1965 are traced back until they reach members of source populations, the expected number of genealogical lines terminating with African individuals is 314, and the expected number terminating in Europeans is 51. Across discrete generations, the peak number of African genealogical ancestors occurs for birth cohorts from the early 1700s. The probability exceeds 50% that at least one European ancestor was born more recently than 1835. Our genealogical perspective can contribute to further understanding the admixture processes that underlie admixed populations. For African Americans, the results provide insight both on how many of the ancestors of a typical African American might have been forcibly displaced in the Transatlantic Slave Trade and on how many separate European admixture events might exist in a typical African-American genealogy.

q-bio.PE↗

arXiv2021

Enumeration of binary trees compatible with a perfect phylogeny

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy--a rooted, binary, timed tree, with tips corresponding to individual sequences. Under the infinitely-many-sites mutation model, mutations are randomly superimposed along the branches of the genealogy, so that every mutation occurs at a chromosomal site that has not previously mutated; if a mutation occurs at an interior branch, then all individuals descending from that branch carry the mutation. The implication is that observed patterns of molecular variation from this model impose combinatorial constraints on the hidden state space of genealogies. In particular, observed molecular variation can be represented in the form of a perfect phylogeny, a tree structure that fully encodes the mutational differences among sequences. For a sample of n sequences, a perfect phylogeny might not possess n distinct leaves, and hence might be compatible with many possible binary tree structures that could describe the evolutionary relationships among the n sequences. Here, we investigate enumerative properties of the set of binary ranked and unranked tree shapes that are compatible with a perfect phylogeny, and hence, the binary ranked and unranked tree shapes conditioned on an observed pattern of mutations under the infinitely-many-sites mutation model. We provide a recursive enumeration of these shapes. We consider both perfect phylogenies that can be represented as binary and those that are multifurcating. The results have implications for computational aspects of the statistical inference of evolutionary parameters that underlie sets of molecular sequences.

q-bio.PE↗

arXiv2020

The distributions under two species-tree models of the number of root ancestral configurations for matching gene trees and species trees

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.

math.CO↗

arXiv2019

Probabilities of unranked and ranked anomaly zones under birth-death models

A labeled gene tree topology that is more probable than the labeled gene tree topology matching a species tree is called \textit{anomalous}. Species trees that can generate such anomalous gene trees are said to be in the \textit{anomaly zone}. Here, probabilities of \textit{unranked} and \textit{ranked} gene tree topologies under the multispecies coalescent are considered. A ranked tree depicts not only the topological relationship among gene lineages, as an unranked tree does, but also the sequence in which the lineages coalesce. In this article, we study how the parameters of a species tree simulated under a constant rate birth-death process can affect the probability that the species tree lies in the anomaly zone. We find that with more than five taxa, it is possible for species trees have both AGTs and ARGTs. The probability of being in either type of anomaly zones increases with more taxa. The probability of AGTs also increases with higher speciation rates. We observe that the probabilities of unranked anomaly zones are higher and grow much faster than those of ranked anomaly zones as the speciation rate increases. Our simulation shows that the most probable ranked gene tree is likely to have the same unranked topology as the species tree. We design the software {\it PRANC} which computes probabilities of ranked gene tree topologies given a species tree under the coalescent model.

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