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Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…

Temporal networks representing a stream of timestamped edges are seemingly ubiquitous in the real-world. However, the massive size and continuous nature of these networks make them fundamentally challenging to analyze and leverage for…

We prove that given a discrete space with $n$ points which is either embedded in a system of $k$ trees, or the Cartesian product of $k$ trees, we can compute all eccentricities in ${\cal O}(2^{{\cal O}(k\log{k})}(N+n)^{1+o(1)})$ time, where…

Regression trees and their ensemble methods are popular methods for nonparametric regression: they combine strong predictive performance with interpretable estimators. To improve their utility for locally smooth response surfaces, we study…

Considering the worst-case scenario, junction tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a…

This paper studies increasing trees on $n$ labeled vertices, in which labels increase from the root to the leaves. It is known that the number of binary increasing trees coincides with the number of alternating permutations (Euler numbers).…

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the…

Generative networks have made it possible to generate meaningful signals such as images and texts from simple noise. Recently, generative methods based on GAN and VAE were developed for graphs and graph signals. However, the mathematical…

We study the size and structure of the largest common subtree (LCS) between two independent Bienaym\'e trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+\kappa)$th moment respectively for some…

We introduce a class of quantum non-Markovian processes -- dubbed process trees -- that exhibit polynomially decaying temporal correlations and memory distributed across time scales. This class of processes is described by a tensor network…

We present the first sublinear-in-$n$ round algorithm for sampling an approximately uniform spanning tree of an $n$-vertex graph in the CongestedClique model of distributed computing. In particular, our algorithm requires…

The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type…

Real-valued time series are ubiquitous in the sciences and engineering. In this work, a general, hierarchical Bayesian modelling framework is developed for building mixture models for times series. This development is based, in part, on the…

We prime-encode the natural numbers via recursive factorisation, iterated to the exponents, generating a corpus of planar rooted trees equivalently represented as Dyck words. This forms a deterministic text endowed with internal rules.…

We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family $\mathcal{D}(n)$, for $n \ge 3$ an integer, and a continuous family $\mathcal{C}(t)$, for $t \ne 1$ a complex number. The construction…

We develop a new Bayesian modelling framework for the class of higher-order, variable-memory Markov chains, and introduce an associated collection of methodological tools for exact inference with discrete time series. We show that a version…

A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using…

Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…

We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple…

The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with…