Related papers: Threshold phenomena for random discrete structures

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more…

Let $G_1,\dots, G_m$ be independent identically distributed Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having vertex sets of random sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and random edge densities $Q_1,\dots, Q_m\in…

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced…

The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size (k), from a fixed set of size (n (n > 2k)), then the largest possible pairwise intersecting family has size (t ={n-1\choose k-1}). We…

In 2009, two different groups independently explored the behavior of random threshold graphs. Here, we extend their techniques to find the distribution of other properties, including matching number, degeneracy, and length of the longest…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…

We propose a method to associate a differentiable Riemannian manifold to a generic many degrees of freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical Statistical Mechanics, we introduce…

For fixed $r\geq 2$, we consider bootstrap percolation with threshold $r$ on the Erd\H{o}s-R\'enyi graph ${\cal G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ which can infect the…

We consider events over the probability space generated by the degree sequences of multiple independent Erd\H{o}s-R\'enyi random graphs, and consider an approximation probability space where such degree sequences are deemed to be sequences…

Let $h>w>0$ be two fixed integers. Let $\orH$ be a random hypergraph whose hyperedges are all of cardinality $h$. To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its vertices positive signs with respect to the hyperedge, and the…

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove…

We model the transmission of information of a message on the Erd\"os-R\'eny random graph with parameters $(n,p)$ and limited resources. The vertices of the graph represent servers that may broadcast a message at random. Each server has a…

We prove a moderate deviation principle for subgraph count statistics of Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an…

We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$…

Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we…

We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which…

In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are…

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…