Related papers: $k$-cut model for the Brownian Continuum Random Tr…
We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process $\{{\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous…
The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some…
One of the fundamental mathematical models for studying random genetic drift is the Kimura equation, derived as the large-population limit of the discrete Wright-Fisher model. However, due to the degeneracy of the diffusion coefficient, it…
The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a Hard-core Random Network model…
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,…
Take a continuous-time Galton-Watson tree and pick $k$ distinct particles uniformly from those alive at a time $T$. What does their genealogical tree look like? The case $k=2$ has been studied by several authors, and the near-critical…
We consider the total cost of cutting down a random rooted tree chosen from a family of so-called very simple trees (which include ordered trees, $d$-ary trees, and Cayley trees); these form a subfamily of simply generated trees. At each…
We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the L\'{e}vy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate…
The $\gamma$-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $\gamma$-stable domain, $\gamma \in (1, 2]$. They form a specific class of…
It is well known that finding extremal values and structures can be hard in weighted graphs. However, if the weights are random, this problem can become way easier. In this paper, we examine the minimal weight of a union of $k$…
Consider symmetric simple exclusion processes, with or without Glauber dynamics on the boundary set, on a sequence of connected unweighted graphs $G_N=(V_N,E_N)$ which converge geometrically and spectrally to a compact connected metric…
This paper studies the large sample asymptotics of data analysis procedures based on the optimization of functionals defined on $k$-NN graphs on point clouds. The paper is framed in the context of minimization of balanced cut functionals,…
We propose discrete random-field models that are based on random partitions of $\mathbb{N}^2$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established…
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$.…
We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random…
Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees…
In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure…
We compute bounds in the expected Ollivier curvature for the Brownian continuum random tree $\mathcal{T}_{\mathbb{e}}$. The results indicate that when the scale dependence of the Ollivier curvature is properly taken into account, the…
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…
Bayesian Additive Regression Trees [BART, Chipman et al., 2010] have gained significant popularity due to their remarkable predictive performance and ability to quantify uncertainty. However, standard decision tree models rely on recursive…