Related papers: Fine asymptotics for the maximum degree in weighte…
We propose a method for cutting down a random recursive tree that focuses on its higher degree vertices. Enumerate the vertices of a random recursive tree of size $n$ according to a decreasing order of their degrees; namely, let…
We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings…
In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large…
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…
We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…
The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic…
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an…
Decision tree learning is increasingly being used for pointwise inference. Important applications include causal heterogenous treatment effects and dynamic policy decisions, as well as conditional quantile regression and design of…
We propose a geometric growth model for weighted scale-free networks, which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks, which are partially determined by the parameters. Analytical…
A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…
We study the growth of a time-ordered rooted tree by probabilistic attachment of new vertices to leaves. We construct a likelihood function of the leaves based on the connectivity of the tree. We take such connectivity to be induced by the…
We examine a random model consisting of objects with positive weights and evolving in discrete time steps, which generalizes certain random graph models. We prove almost sure convergence for the weight distribution and show scale-free…
We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z^{(\mu)}_N$,…
We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile…
We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and…
We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results…
In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is…
We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16].…