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For the size of the largest component in a supercritical random geometric graph, this paper estimates its expectation which tends to a polynomial on a rate of exponential decay, and sharpens its asymptotic result with a central limit…

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition,…

We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes…

We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of…

We present a new notion of limits of weighted directed graphs of growing size based on convergence of their random quotients. These limits are specified in terms of random exchangeable measures on the unit square. We call our limits…

Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n)…

Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We…

A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$ vertices which begins with the edgeless graph, and where at each step a single edge is added according to a uniform distribution on the missing edges. It is well known…

Random graph models are playing an increasingly important role in various fields ranging from social networks, telecommunication systems, to physiologic and biological networks. Within this landscape, the random Kronecker graph model,…

We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyse the limit distribution of the degree of a…

For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider…

This Letter introduces a generalization of known duplication-divergence models for growing random graphs. This general duplication-divergence model includes a new coupled divergence asymmetry rate, which allows to obtain the structure of…

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As…

In this paper, we consider a one-dimensional random geometric graph process with the inter-nodal gaps evolving according to an exponential AR(1) process, which may serve as a mobile wireless network model. The transition probability matrix…

Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs.

We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent $2<\gamma<3$. Degree heterogeneity increases the presence of triangles in…

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…

Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of…