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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…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based…
A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information…
Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many…
Graphs are complex objects that do not lend themselves easily to typical learning tasks. Recently, a range of approaches based on graph kernels or graph neural networks have been developed for graph classification and for representation…
Graphical domination was first introduced in [1] in the context of combinatorial threshold-linear networks (CTLNs). There it was shown that when a domination relationship exists between a pair of vertices in a graph, certain fixed points in…
Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work…
Neural network models in neuroscience allow one to study how the connections between neurons shape the activity of neural circuits in the brain. In this chapter, we study Combinatorial Threshold-Linear Networks (CTLNs) in order to…
We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It…
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay…
In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous…
We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both…
We define notions of local topological convergence and local geometric convergence for embedded graphs in $\mathbb{R}^n,$ and study their properties. The former is related to Benjamini-Schramm convergence, and the latter to weak convergence…
We propose a general class of co-evolving tree network models driven by local exploration where new vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the…
Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a…
Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global…
In this article, we study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of…
This paper addresses the problem of online network topology inference for expanding graphs from a stream of spatiotemporal signals. Online algorithms for dynamic graph learning are crucial in delay-sensitive applications or when changes in…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…