Hypergraph Change Point Detection using Adapted Cardinality-Based Gadgets: Applications in Dynamic Legal Structures

TL;DR

This paper introduces a novel hypergraph transformation method using adapted cardinality-based gadgets and Laplacian analysis, enabling effective change point detection in dynamic systems, including legal hypergraphs.

Contribution

The paper presents a new hypergraph-to-graph transformation technique that preserves spectral properties, improving change point detection in dynamic hypergraphs, especially in legal data applications.

Findings

Outperforms clique and star expansions in spectral preservation

Effective change point detection demonstrated on synthetic and real datasets

Applied successfully to analyze dynamic legal hypergraphs from court data

Abstract

Hypergraphs provide a robust framework for modeling complex systems with higher-order interactions. However, analyzing them in dynamic settings presents significant computational challenges. To address this, we introduce a novel method that adapts the cardinality-based gadget to convert hypergraphs into strongly connected weighted directed graphs, complemented by a symmetrized combinatorial Laplacian. We demonstrate that the harmonic mean of the conductance and edge expansion of the original hypergraph can be upper-bounded by the conductance of the transformed directed graph, effectively preserving crucial cut information. Additionally, we analyze how the resulting Laplacian relates to that derived from the star expansion. Our approach was validated through change point detection experiments on both synthetic and real datasets, showing superior performance over clique and star expansions in maintaining spectral information in dynamic settings. Finally, we applied our method to analyze a dynamic legal hypergraph constructed from extensive United States court opinion data.