Related papers: Tree split probabilities determine the branch leng…

We examine a discrete random recursive tree growth process that, at each time step, either adds or deletes a node from the tree with probability $p$ and $1-p$, respectively. Node addition follows the usual uniform attachment model. For node…

The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical…

We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A…

With advances in sequencing technologies, there are now massive amounts of genomic data from across all life, leading to the possibility that a robust Tree of Life can be constructed. However, "gene tree heterogeneity", which is when…

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event…

We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability…

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…

Markov chains are convenient means of generating realizations of networks with a given (joint or otherwise) degree distribution, since they simply require a procedure for rewiring edges. The major challenge is to find the right number of…

Kirchhoff's matrix tree theorem is a well-known result that gives a formula for the number of spanning trees in a finite, connected graph in terms of the graph Laplacian matrix. A closely related result is Wilson's algorithm for putting the…

We introduce a Markov Chain Monte Carlo algorithm which samples from the space of spanning trees of complete graphs using local rewiring operations only. The probability distribution of graphs of this kind is shown to depend on the…

Recently there has been renewed interest in phylogenetic inference methods based on phylogenetic invariants, alongside the related Markov invariants. Broadly speaking, both these approaches give rise to polynomial functions of sequence site…

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies…

Phylogenetics uses alignments of molecular sequence data to learn about evolutionary trees. Substitutions in sequences are modelled through a continuous-time Markov process, characterised by an instantaneous rate matrix, which standard…

For different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies…

We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root,…

We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) 409-432]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion…

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the…

Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following…

Markov chain Monte Carlo algorithms play a key role in the Bayesian approach to phylogenetic inference. In this paper, we present the first theoretical work analyzing the rate of convergence of several Markov chains widely used in…

Ancestral maximum likelihood (AML) is a method that simultaneously reconstructs a phylogenetic tree and ancestral sequences from extant data (sequences at the leaves). The tree and ancestral sequences maximize the probability of observing…