Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

TL;DR

This paper introduces a new Cheeger constant combining frustration index and expansion rate, establishing inequalities for magnetic Laplacians on graphs and manifolds, and develops spectral clustering algorithms for complex graph structures.

Contribution

It presents novel Cheeger inequalities for magnetic Laplacians and introduces spectral clustering methods based on advanced geometric metrics.

Findings

Proved Cheeger and higher order Cheeger inequalities for magnetic Laplacians.

Developed spectral clustering algorithms for partially oriented graphs.

Unified perspective on structural balance theory and gauge invariance.

Abstract

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary's structural balance theory of signed graphs and the gauge invariance of magnetic potentials.