Random Connection Hypergraphs

TL;DR

The paper introduces the Random Connection Hypergraph Model (RCHM), a weighted random connection framework built on a bipartite Poisson system to model higher-order interactions via the Dowker complex. It shows how simple connectivity rules $F(p,p') = |x-z| u^{\gamma} w^{\gamma'}$ yield scale-free higher-order degrees and enables rigorous limit theorems for simplex counts and Betti numbers, with normal and stable regimes governed by tail indices $\gamma$ and $\gamma'$. The authors prove finite intensities, Palm distribution formulas, and both upper and lower tails for simplex degrees, establish CLTs for Betti numbers under light tails, and derive normal/stable limits for edge and simplex counts; they complement theory with extensive simulations and an arXiv-based collaboration network study. The results provide a tractable, geometry-infused, scale-free model for higher-order networks and offer practical tools for hypothesis testing and model-data comparisons in real-world bipartite systems. Overall, the work advances understanding of how geometry, weights, and higher-order topology interact in random hypergraphs and demonstrates applicability to large-scale collaboration data.

Abstract

In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes, and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of various graph-theoretic and topological characteristics, including higher-order degree distributions, Betti numbers of the associated Dowker complex, and simplex counts. In particular, for the latter quantity, we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation with a simulation study and an application to the collaboration network extracted from the arXiv dataset.

Figures (11)

• Figure 1: Partition of $\S$ into rectangles.
• Figure 2: Main properties of a network sample generated by our model
• Figure 3: Fluctuation of the degree distribution exponents for different network sizes. The top row shows the power-law exponents when estimated from values larger than $\xmin = 10$. In contrast, the bottom row contains boxplots for the distribution of the exponents estimated from values larger than $\xmin = 15$.
• Figure 4: Distribution of the first Betti numbers with different $\g$ and $\g'$ parameters
• Figure 5: Distribution of edge counts with different $\g$ and $\g'$ parameters. For each combination of $\g$ and $\g'$, we show the distribution of edge counts with the fitted normal distribution and the corresponding Q-Q plot.

Theorems & Definitions (24)

• Proposition 2.1: Finiteness of the $m$-simplex-intensity
• Proposition 2.2: Distribution of the typical $m$-simplex
• Theorem 2.3: Scale-freeness of higher-order degrees
• Theorem 2.4: Asymptotic normality of Betti numbers
• Theorem 2.5: Normal and stable limits of edge counts
• Theorem 2.6: Normal and stable limits of simplex counts
• proof : Proof of \ref{['prop:finite_intensity']}
• proof : Proof of \ref{['prop:palm_distribution']}
• Lemma 4.1: Pairwise intersections
• proof