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

Hiroki Matsumoto, Takahiro Yoshida, Ryoma Kondo, Ryohei Hisano

TL;DR

A novel method is introduced that adapts the cardinality-based gadget to convert hypergraphs into strongly connected weighted directed graphs, complemented by a symmetrized combinatorial Laplacian, and demonstrates 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.

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.

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

TL;DR

A novel method is introduced that adapts the cardinality-based gadget to convert hypergraphs into strongly connected weighted directed graphs, complemented by a symmetrized combinatorial Laplacian, and demonstrates 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.

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.
Paper Structure (13 sections, 2 theorems, 19 equations, 3 figures, 1 table)

This paper contains 13 sections, 2 theorems, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Hypergraph Conductance and Graph Reduction
  4. Comparison Between Our Approach and Star Expansion
  5. Laplacian Anomaly Detection
  6. Data
  7. Synthetic Dataset with Ground-Truth Change Points
  8. Dynamic Legal Hypergraphs
  9. Results
  10. Synthetic Dataset
  11. Dynamic Legal Hypergraph
  12. Conclusion
  13. Acknowledgment

Key Result

proposition thmcounterproposition

If $\mathrm{vol}_{\mathcal{H}}(S)\leq \mathrm{vol}_{\mathcal{H}}(V\backslash S)$ and $|S| \leq |V\backslash S|$, then $\mu_{\mathcal{H}}(S)$ equals the harmonic mean of $\phi_{\mathcal{H}}(S)$ and $\frac{1}{\beta}\psi_{\mathcal{H}}(S)$.

Figures (3)

  • Figure 1: Time series plot depicting the share over time.
  • Figure 2: Estimated performance of three reduction techniques on a synthetic dataset. The top $3\%$ of anomalous time points are marked by 'X,' and the vertical dashed line indicates the ground truth change points.
  • Figure 3: Results for the real data. The top $5\%$ of anomalous time points are marked by 'X,'

Theorems & Definitions (5)

  • definition thmcounterdefinition: Adapted CB-gadget
  • proposition thmcounterproposition
  • proof
  • theorem thmcountertheorem
  • proof