Spectral methods reveal new insights into hypergraph structure and expansion
A Cheeger Inequality for Hypergraphs and Its Applications
Data Structures and Algorithms
Summary
Hypergraphs are ways to describe connections involving many things at once, not just pairs. The authors created new mathematical tools to analyze these complex connections and showed a special inequality that relates how connected the hypergraph is to numbers derived from its structure. They confirmed their findings work well by testing on known shapes and extended the results to more complex cases. Their work also introduced new examples of hypergraphs with optimal connectivity properties.
What this means in practice
- •For data scientists: Use improved spectral tools to detect meaningful groups in data with complex multi-way relationships.
- •For network engineers: Design hypergraph-based network models with guaranteed connectivity and expansion properties for efficient communication and robustness.
A theory result. No direct application yet.
Authors
Raj Kamal, Amitabha Bagchi
Abstract
Hypergraphs provide a natural framework for modeling higher-order relationships, but the development of spectral techniques with provable guarantees for general non-uniform hypergraphs remains challenging. Building on Banerjee's normalized adjacency matrix and Spiro's averaging-based diffusion framework, we develop a spectral framework for non-uniform hypergraphs and establish Cheeger's inequality for their conductance. A fundamental result in the spectral theory of hypergraphs asserts that, for every non-covering hypergraph, the second-smallest eigenvalue of its normalized Laplacian is at most one. This spectral characterization yields an improved Cheeger's inequality for non-covering hypergraphs, and we show that the resulting inequality is tight on both sides using cycle and cube hypergraphs. Our framework further yields higher-order Cheeger inequalities and provides theoretical guarantees for Fiedler's spectral partitioning algorithm, all in the setting of hypergraphs. Finally and most notably, we construct a new family of optimal hypergraph expanders that is tight for the Alon--Boppana bound.