Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

Lange, Carsten; Liu, Shiping; Peyerimhoff, Norbert; Post, Olaf

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.

Citation

Lange, C., Liu, S., Peyerimhoff, N., & Post, O. (2015). Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians. Calculus of Variations and Partial Differential Equations, 54(4), 4165-4196. https://doi.org/10.1007/s00526-015-0935-x

• Journal Article Type: Article
• Acceptance Date: Sep 14, 2015
• Online Publication Date: Nov 5, 2015
• Publication Date: Dec 1, 2015
• Deposit Date: Aug 14, 2015
• Publicly Available Date: Dec 1, 2015
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 54
• Issue: 4
• Pages: 4165-4196
• DOI: https://doi.org/10.1007/s00526-015-0935-x
• Public URL: https://durham-repository.worktribe.com/output/1433793