Conference Agenda

In this talk, a novel random hypergraph model is introduced. In accordance to the standard theory, the hypergraphs are represented as bipartite graphs, with both vertex sets derived from marked Poisson point processes. After establishing the model, we investigate the limit theory of a variety of characteristics, including higher-order degree distributions, Betti numbers, and simplex counts. The theoretical results will be complemented by a simulation study to analyze finite-size effects to help understanding of the model's behavior in practical scenarios. Finally, to demonstrate the real-world applicability of the results, we examine how this model can be employed to understand scientific collaboration networks extracted from arXiv.

The theory of graph limits considers the convergence of sequences of graphs with a divergent number of vertices. From an applied perspective, it aims at the convenient representation of large networks. In this talk, I will give a brief introduction to graph limits and report on recent extensions to weighted graphs and more general combinatorial objects as hypergraphs. In particular, I will develop the theory of probability graphons, focusing on the right convergence viewpoint, and the equivalent notion of P-variables convergence. The relation between probability graphons and P-variables is analogous to the relation between probability measures and random variables. I will also explain how these notions can be generalised to hypergraph limits and how they relate to many other areas of mathematics, statistics and applications.

The aim of this work is to establish quantitative multivariate central limit theorems with respect to the convex distance for a large class of geometric functionals of marked binomial point processes. More precisely, the underlying functionals are assumed to be sums of exponentially stabilizing score functions which additionally have to satisfy certain moment conditions. The found results are also illustrated by some specific examples from stochastic geometry. The whole framework and results, which profoundly make use of the findings in Kasprzak and Peccati (Ann. Appl. Probab. 33 (2023), 3449-3492) and Lachieze-Rey, Schulte and Yukich (Ann. Appl. Probab. 29 (2019), 931-993), can be regarded as binomial counterparts of those in Schulte and Yukich (Ann. Appl. Probab. 33 (2023), 507-548), which were concerned with Poisson point processes.

The $\beta$-Delaunay and $\beta^\prime$-Delaunay triangulations are models generalizing classical Poisson-Delaunay triangulation on the Euclidean space. Both were introduced recently by Gusakova, Kabluchko, and Thäle in a series of papers. We investigate the distribution of a maximal vertex degree in a growing window for these models. For $\beta$-Delaunay we show that it behaves similarly to the classical case, that is, concentrates on a finite number of values (just two values if we are on a plane). On the other hand, for $\beta^\prime$-Delaunay we show that the situation is completely different.