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

Higher-order networks, which model interactions among groups of entities of an arbitrary number, emerged as a generalization of simple networks with pairwise interactions. Their analysis requires the development of stochastic models that can capture complex spatial, topological, and temporal dependencies. This thesis presents new probabilistic frameworks for modeling such networks grounded in point process theory.<br/><br/>The first contribution establishes a Poisson approximation result for the number of degree-k nodes in weighted random connection models. Nodes are embedded in Rd via a weighted Poisson point process, and edges are formed based on both spatial proximity and node weights. We identify scaling regimes under which the spatial distribution of degree-k nodes converges in Kantorovich–Rubinstein distance to a homogeneous Poisson point process.<br/><br/>In the second part, we analyze the age-dependent random connection model, a spatial network 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 how well the model captures the topological features of the network.<br/><br/>Next, the framework is extended to hypergraphs. Both network nodes and hyperedges are modeled as weighted Poisson point processes, and hyperedges are formed based on joint connections to points representing the hyperedges. In this model, we prove normal and stable limit theorems for simplex counts, Betti numbers, and edge statistics. We also present a simulation study and an application to a collaboration network extracted from the arXiv dataset, comparing the model with real-world hypergraphs.<br/><br/>Finally, a dynamic version of the hypergraph model is proposed, where we equip the vertices with birth-death dynamics. We establish two functional limit theorems for the edge-count process in the model: for light-tailed degree distributions, it converges<br/>to a Gaussian process with Matérn-type covariance. In heavy-tailed regimes, the edge count process converges to a non-Markovian, non-Lévy stable process. These results constitute the first dynamic limit theorems for spatial higher-order networks.

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