PhD Thesis: Stochastic Models of Higher-Order Networks - Point Processes and Topological Data Analysis

Higher-order networks model interactions among groups of entities of an arbitrary number. This thesis introduces stochastic models grounded in point process theory that are capable of capturing complex spatial, topological, and temporal dependencies in higher-order networks. The first contribution examines weighted random connection models in which nodes are represented as a weighted Poisson point process, and edges are formed depending on spatial proximity and node weights. We show that the spatial distribution of degree-k nodes converges to a homogeneous Poisson point process. In the second part, we analyze the age-dependent random connection model viewed as a higher-order network. Limit theorems are shown for higher-order degree distributions, Betti numbers, and edge counts. Then, the model is fitted to a real-world collaboration network, and hypothesis tests are conducted to assess the model against the dataset. Next, the framework is extended to hypergraphs, where both network nodes and hyperedges are modeled as Poisson point processes. In this model, we prove normal and stable limit theorems for simplex counts, Betti numbers, and edge statistics. We also present an application to a collaboration network to compare the model with real-world hypergraphs. Finally, a dynamic version of the hypergraph model is proposed, where we equip the vertices with birth-death dynamics. We establish a normal and a stable functional limit theorem for the edge-count process in the model. These results constitute the first functional limit theorems for spatial higher-order networks.