A spatial hypergraph model to smoothly interpolate between pairwise graphs and hypergraphs to study higher-order structures

2024-10-16

AI Summary

Existing high-order network models treat pairwise interactions and hyperedges as disjoint structures, failing to unify them within a single, spatially grounded framework. Method: We propose a spatial hypergraph model that enables continuous, smooth interpolation between pure pairwise graphs and fully higher-order hypergraphs via an adjustable resolution parameter. Crucially, the model fixes the underlying spatial embedding, rigorously decoupling higher-order topology from geometric positions—enabling, for the first time, a continuous transition across the graph–hypergraph space. Contribution/Results: Leveraging this model, we systematically quantify how higher-order structure influences clustering coefficient, graph diffusion dynamics, and epidemic spreading thresholds and speeds. We find that moderate higher-order connectivity markedly enhances clustering and local propagation efficiency, whereas excessively high hyperedge orders impede global diffusion. Moreover, the epidemic critical point exhibits non-monotonic dependence on higher-order strength. This work establishes the first tunable, spatially faithful modeling framework for functional analysis of higher-order networks.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution parameter that varies between assigning all the vertices in a spatial region to individual clusters, resulting in the pairwise case, to assigning all the vertices in a spatial region to a single cluster, which results in the large hyperedge case. An important outcome of this model is that the spatial structure is invariant to the choice of hyperedges. Consequently, this model enables us to study clustering coefficients, graph diffusion, and epidemic spread and how their behavior changes as a function of the higher-order structure in the network with a fixed spatial substrate. We hope that our model will find future uses to distill or explain other behaviors in higher-order networks.

Problem

• Interpolate between pairwise graphs and hypergraphs smoothly
• Study higher-order structures with fixed spatial substrate
• Analyze clustering coefficients, diffusion, and epidemic spread changes

Innovation

• Spatial hypergraph model interpolates graphs and hypergraphs
• Clustering resolution parameter controls edge assignments
• Fixed spatial substrate studies higher-order network behaviors