Related papers: Sharp threshold functions for the random intersect…

The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily…

In many areas such as computational biology, finance or social sciences, knowledge of an underlying graph explaining the interactions between agents is of paramount importance but still challenging. Considering that these interactions may…

For a given graph $F$ we consider the family of (finite) graphs $G$ with the Ramsey property for $F$, that is the set of such graphs $G$ with the property that every two-colouring of the edges of $G$ yields a monochromatic copy of $F$. For…

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk…

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we…

We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces.

Sampling random graphs is essential in many applications, and often algorithms use Markov chain Monte Carlo methods to sample uniformly from the space of graphs. However, often there is a need to sample graphs with some property that we are…

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In…

High-dimensional feature selection is a central problem in a variety of application domains such as machine learning, image analysis, and genomics. In this paper, we propose graph-based tests as a useful basis for feature selection. We…

Consider a random graph G in G(n,p) and the graph property: G contains a copy of a specific graph H. (Note: H depends on n; a motivating example: H is a Hamiltonian cycle.) Let q be the minimal value for which the expected number of copies…

Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on…

In this paper, we develop a novel paradigm, namely hypergraph shift, to find robust graph modes by probabilistic voting strategy, which are semantically sound besides the self-cohesiveness requirement in forming graph modes. Unlike the…

We study Hamilton cycles and perfect matchings in a uniform attachment graph. In this random graph, vertices are added sequentially, and when a vertex $t$ is created, it makes $k$ independent and uniform choices from $\{1,\dots,t-1\}$ and…

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a…

We determine the \emph{exact} minimum $\ell$-degree threshold for perfect matchings in $k$-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than $\frac{1}{2} \binom{n}{k- \ell}$.…

In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. We prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We…

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$.…

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with…

We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed…

A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with…