Related papers: The Uncover Process for Random Labeled Trees
We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common leaves have edge disjoint tree realizations. This conjecture is known to be true for $2$ and…
Suppose that a cascade (e.g., an epidemic) spreads on an unknown graph, and only the infection times of vertices are observed. What can be learned about the graph from the infection times caused by multiple distinct cascades? Most of the…
This work introduces two techniques for the design and analysis of branching algorithms, illustrated through the case study of the Vertex Cover problem. First, we present a method for automatically generating branching rules through a…
We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge-colouring of the…
We present a graph theoretical approach to the configurational statistics of random tree-like objects, such as randomly branching polymers. In particular, for ideal trees we show that Pr\"ufer labelling provides: (i) direct access to the…
A learning algorithm is presented which given the structure of a causal tree, will estimate its link probabilities by sequential measurements on the leaves only. Internal nodes of the tree represent conceptual (hidden) variables…
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…
Given a solution to a recursive distributional equation, a natural (and non-trivial) question is whether the corresponding recursive tree process is endogenous. That is, whether the random environment almost surely defines the tree process.…
For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform…
We define a mapping from transition-based parsing algorithms that read sentences from left to right to sequence labeling encodings of syntactic trees. This not only establishes a theoretical relation between transition-based parsing and…
The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise…
We propose a principled method for autoencoding with random forests. Our strategy builds on foundational results from nonparametric statistics and spectral graph theory to learn a low-dimensional embedding of the model that optimally…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
A layerwise search in a split-by-edges tree (as defined by Br{\ae}ndeland, 2015) of agiven graph produces a maximum independent set in exponential time. A depth-first search produces an independent set, which may or may not be a maximum, in…
Bootstrap percolation is a process that is used to model the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds…
We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving…
We propose a method for the classification of objects that are structured as random trees. Our aim is to model a distribution over the node label assignments in settings where the tree data structure is associated with node attributes…
We study the problem of detecting the edge correlation between two random graphs with $n$ unlabeled nodes. This is formalized as a hypothesis testing problem, where under the null hypothesis, the two graphs are independently generated;…
We extend the latent position random graph model to the line graph of a random graph, which is formed by creating a vertex for each edge in the original random graph, and connecting each pair of edges incident to a common vertex in the…
Random walks are a common model for exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are…