Related papers: Largest connected component in duplication-diverge…
The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…
We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with…
The largest balanced element in signed graphs plays a vital role in helping researchers understand the fundamental structure of the graph, as it reveals valuable information about the complex relationships between vertices in the network.…
Consider a critical random multigraph $\mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second…
Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree $\Delta$. We investigate the relative size of the giant component $S$…
In this paper, we study the level-set of the zero-average Gaussian Free Field on a uniform random $d$-regular graph above an arbitrary level $h\in (-\infty, h_{\star})$, where $h_{\star}$ is the level-set percolation threshold of the GFF on…
Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E}…
Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$…
We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is…
We study the percolation properties of the growing clusters model. In this model, a number of seeds placed on random locations on a lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually…
In complex networks the degrees of adjacent nodes may often appear dependent -- which presents a modelling challenge. We present a working framework for studying networks with an arbitrary joint distribution for the degrees of adjacent…
Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…
It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are…
We consider homological edge percolation on a sequence $(\mathcal{G}_t)_t$ of finite graphs covered by an infinite (quasi)transitive graph $\mathcal{H}$, and weakly convergent to $\mathcal{H}$. Namely, we use the covering maps to classify…
The cut-set $\partial(S)$ of a graph $G=(V,E)$ is the set of edges that have one endpoint in $S\subset V$ and the other endpoint in $V\setminus S$, and whenever $G[S]$ is connected, the cut $[S,V\setminus S]$ of $G$ is called a connected…
We establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency…
We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with $p$ parts generated according to a given degree sequence $n_i^{\mathbf{d}}(n)$ which denotes the number…
We study numerically a model of nonequilibrium networks where nodes and links are added at each time step with aging of nodes and connectivity- and age-dependent attachment of links. By varying the effects of age in the attachment…