Functional limit theorems for edge counts in dynamic random connection hypergraphs

We introduce a dynamic random hypergraph model constructed from a bipartite graph. In this model, the first vertex set consists of vertices, while the second vertex set represents interactions. Both vertices and interactions are generated by marked Poisson point processes and are equipped with scalar weights that influence their connection radii. Additionally, the vertices are assigned independent birth–death processes with exponential lifetimes, whereas the interactions are assigned interaction times. Connections between vertices and interactions are established based on the weights and the temporal dynamics, resulting in 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 increases. In the challenging heavy-tailed regime with infinite variance, we prove convergence to a novel stable process that is neither Lévy nor Markov.