Related papers: Multigraph limit of the dense configuration model …
We propose a wide class of preferential attachment models of random graphs, generalizing previous approaches. Graphs described by these models obey the power-law degree distribution, with the exponent that can be controlled in the models.…
Time-evolving random graph models have appeared and have been studied in various fields of research over the past decades. However, the rigorous mathematical treatment of large graphs and their limits at the process-level is still in its…
For a sequence of random graphs, the limit law we refer to is the existence of a limiting probability of any graph property that can be expressed in terms of predicate logic. A zero-one limit law is shown by Shelah and Spencer for…
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we…
The theory of limits of dense graph sequences was initiated by Lovasz and Szegedy. We give a possible generalization of this theory to multigraphs. Our proofs are based on the correspondence between dense graph limits and countable,…
We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We…
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 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 introduce a broad class of random graph models: the generalised hypergeometric ensemble (GHypEG). This class enables to solve some long standing problems in random graph theory. First, GHypEG provides an elegant and compact formulation…
Random graph null models have found widespread application in diverse research communities analyzing network datasets, including social, information, and economic networks, as well as food webs, protein-protein interactions, and neuronal…
Letting $\mathcal{M}$ denote the space of finite measures on $\mathbb{N}$, and $\mu_\lambda\in\mathcal{M}$ denote the Poisson distribution with parameter $\lambda$, the function $W:[0,1]^2\to\mathcal{M}$ given by \[ W(x,y)=\mu_{c\log x\log…
Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they…
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…
Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration…
Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence $\mathcal{D}$ and with surplus $k$. We call those random graphs…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
Graph convolutional networks (GCNs) are a widely used method for graph representation learning. To elucidate the capabilities and limitations of GCNs, we investigate their power, as a function of their number of layers, to distinguish…
Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…