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We extend the theory of probability graphons, continuum representations of edge-decorated graphs arising in graph limits theory, to the 'right convergence' point of view. First of all, we generalise the notions of overlay functionals and…
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.
The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and…
We consider an SIR-type (Susceptible $\to$ Infected $\to$ Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model…
Graph databases in many applications---semantic web, transport or biological networks among others---are not only large, but also frequently modified. Evaluating graph queries in this dynamic context is a challenging task, as those queries…
In many applications concerning statistical graphical models the data originate from several subpopulations that share similarities but have also significant differences. This raises the question of how to estimate several graphical models…
In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we…
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…
We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block…
Similarity metrics are central in the theory of large networks and graph limits. For bounded-degree graphs, the Benjamini--Schramm metric records the distribution of rooted neighbourhoods, while the stronger colored-neighbourhood metric…
We propose generalizations of a number of standard network models, including the classic random graph, the configuration model, and the stochastic block model, to the case of time-varying networks. We assume that the presence and absence of…
We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.…
We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy…
The success of deep learning has revolutionized many fields of research including areas of computer vision, text and speech processing. Enormous research efforts have led to numerous methods that are capable of efficiently analyzing data,…
We study the challenges of applying deep learning to gene expression data. We find experimentally that there exists non-linear signal in the data, however is it not discovered automatically given the noise and low numbers of samples used in…
Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of…
This paper is concerned with distributed computation of several commonly used centrality measures in complex networks. In particular, we propose deterministic algorithms, which converge in finite time, for the distributed computation of the…
For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours…