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.
Highlights
Cheeger’s inequality is one of the most fundamental and important estimates in spectral geometry. It was first proved by Cheeger for the Laplace-Beltrami operator on a Riemannian manifold [7] and later extended to the setting of discrete graphs, see e.g., [1,2,6,11], demonstrating the close relationship between the spectrum and the geometry of the underlying space
Another example is an improved Cheeger’s inequality for finite graphs by Kwok et al [27], which was subsequently used to establish an optimal dimension-free upper bound of eigenvalue ratios for weighted closed Riemannian manifolds with nonnegative Ricci curvature [33]
We focus on finite graphs and compact Riemannian manifolds in this paper
Summary
Cheeger’s inequality is one of the most fundamental and important estimates in spectral geometry It was first proved by Cheeger for the Laplace-Beltrami operator on a Riemannian manifold [7] and later extended to the setting of discrete graphs, see e.g., [1,2,6,11], demonstrating the close relationship between the spectrum and the geometry of the underlying space. While all operators studied in [3,32] are bounded, we show that finding proper metrics for clustering is useful for unbounded operators: the spectral clustering algorithms via metrics on complex projective spaces are crucial to prove the higher order Cheeger inequalities of the magnetic Laplacian on a closed Riemannian manifold (Lemma 7.8).
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