相关论文: A simple branching process approach to the phase t…

We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the…

We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n,p) in the critical window p=n^{-1}+lambda n^{-4/3}. In particular, we show that this point process has a surprising…

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the…

Consider the complete graph on \(n\) vertices where each edge is independently open with probability \(p,\) or closed otherwise. Phase transitions for such graphs for \(p = \frac{C}{n}\) have previously been studied using techniques like…

Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a random $r$-regular graph where $r\ge 3$ is fixed and $n=|G|\to\infty$. We study the percolation phase transition on such graphs $G$, i.e., the emergence as $p$…

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p)…

The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalisations of the notion of a component, and for all…

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar…

Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph $G(n,p)$ above the phase…

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint…

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…

We study the "rank 1 case" of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result is new, it completes the…

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\{1,\dotsc,n\}$ with $m$ edges…

We provide a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. Our…

The random coloured graph $G_c(n,p)$ is obtained from the Erd\H{o}s-R\'{e}nyi binomial random graph $G(n,p)$ by assigning to each edge a colour from a set of $c$ colours independently and uniformly at random. It is not hard to see that,…

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the…

We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walk, introduced by…

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window…

The general theory of the branching processes is used for establishing the relation between the parameters $k$ and $\bar n$ of the negative binomial distribution. This relation gives the possibility to describe the overall data on…

Consider a continuous-time binary branching process conditioned to have population size n at some time t, and with a chance p for recording each extinct individual in the process. Within the family tree of this process, we consider the…