Related papers: The strong giant in a random digraph

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…

Given disjoint subsets $T_1,\ldots,T_m$ of "not too large" primes up to $x$, we establish that for a random integer $n$ drawn from $[1,x]$, the $m$-dimensional vector enumerating the number of prime factors of $n$ from $T_1,\ldots,T_m$…

We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In…

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…

Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric…

Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $\theta_i$ and scale parameter $\lambda_i$ (resp.…

Consider the following model of strong-majority bootstrap percolation on a graph. Let r be some positive integer, and p in [0,1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every…

We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on…

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations…

We generalize an algorithm used widely in the configuration model such that power-law degree sequences with the degree exponent $\lambda$ and the number of links per node $K$ controllable independently may be generated. It yields the degree…

This paper studies networks where all nodes are distributed on a unit square $A\triangleq[(-1/2,1/2)^{2}$ following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly…

Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…

Let $\tau$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study…

The model of heavy Wigner matrices generalizes the classical ensemble of Wigner matrices: the sub-diagonal entries are independent, identically distributed along to and out of the diagonal, and the moments its entries are of order 1/N,…

For a permuton $\mu$ let $H_n(\mu)$ denote the Shannon entropy of the sampling distribution of $\mu$ on $n$ points. We investigate the asymptotic growth of $H_n(\mu)$ for a wide class of permutons. We prove that if $\mu$ has a non-vanishing…

We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. The edge probabilities are moderated by vertex weights, and are such that the degree of…

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied.…

We study the size of the largest biconnected components in sparse Erd\H{o}s-R\'enyi graphs with finite connectivity and Barab\'asi-Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we…

In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…

We offer a solution to a long-standing problem in the physics of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity -- the propensity for two neighbors of a network node also to be…