Related papers: An example of graph limits of growing sequences of…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…
A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…
Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and…
In this note we make some specific observations on the distribution of the degree of a given vertex in certain model of randomly growing networks. The rule for network growth is the following. Starting with an initial graph of minimum…
A graph generative model defines a distribution over graphs. One type of generative model is constructed by autoregressive neural networks, which sequentially add nodes and edges to generate a graph. However, the likelihood of a graph under…
We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number,…
We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random…
In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform…
A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in…
We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree…
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…
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
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…
Consider a random graph process where vertices are chosen from the interval $[0,1]$, and edges are chosen independently at random, but so that, for a given vertex $x$, the probability that there is an edge to a vertex $y$ decreases as the…
We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs,…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…