Related papers: The height of random $k$-trees and related branchi…
Our goal is to quickly find top $k$ lists of nodes with the largest degrees in large complex networks. If the adjacency list of the network is known (not often the case in complex networks), a deterministic algorithm to find a node with the…
We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the…
We code Galton-Walton trees by a continuous height process, in order to give a precise meaning to the convergence of forests of trees. This allows us to establish the convergence of the forest of genealogical trees of the branching process…
We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random…
This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a…
This paper derives the asymptotic distribution of the normalized $k$-th maximum order statistics of a sequence of non-central chi-square random variables with non-identical non-centrality parameter. We demonstrate the utility of these…
We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time…
We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm…
We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice on the average height of planted plane trees,…
Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a…
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…
We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the…
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…
Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…
Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of $k$ paths ($k$ is odd) along branches of trees or, equivalently, $k$ loop--erased random walks. Starting and ending points of…
It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the…
The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this…
The connection between the maximum spanning tree in a directed graph and the best dependency tree of a sentence has been exploited by the NLP community. However, for many dependency parsing schemes, an important detail of this approach is…
In this paper, we provide algorithms to rank and unrank certain degree-restricted classes of Cayley trees (spanning trees of the n-vertex complete graph). Specifically, we consider classes of trees that have a given set of leaves or a fixed…
We provide asymptotics for the range R(n) of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that R(n)/n converges to a…