Functional limit theorems for edge counts in dynamic random connection hypergraphs

TL;DR

This paper introduces a dynamic hypergraph model based on bipartite graphs with Poisson processes, analyzing edge count fluctuations and establishing limit theorems for both finite and heavy-tailed regimes.

Contribution

It presents a new dynamic hypergraph model with power-law degrees and proves functional limit theorems for edge counts in different tail regimes, including a novel stable process.

Findings

Gaussian AR(2)-type process in finite fourth moment regime

Convergence to a non-Markov stable process in heavy-tailed regime

Model captures power-law degree distributions in dynamic hypergraphs

Abstract

We introduce a dynamic random hypergraph model constructed from a bipartite graph. In this model, both vertex sets of the bipartite graph are generated by marked Poisson point processes. Vertices of both vertex sets are equipped with marks representing their weight that influence their connection radii. Additionally, we also assign the vertices of the first vertex set a birth-death process with exponential lifetimes and the vertices of the second vertex set a time instant representing the occurrence of the corresponding vertices. Connections between vertices are established based on the marks and the birth-death processes, leading to a weighted dynamic hypergraph model featuring power-law degree distributions. We analyze the edge-count process in two distinct regimes. In the case of finite fourth moments, we establish a functional central limit theorem for the normalized edge count, showing convergence to a Gaussian AR(2)-type process as the observation window grows. In the challenging case of the heavy-tailed regime with infinite variance, we prove convergence to a novel stable process that is not L\'evy and not even Markov.