Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

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.

Magnetic Schrödinger operators on periodic discrete graphs

“…In this section we deal with a more general class of magnetic Laplacians. These generalized magnetic Laplacians on finite and infinite graphs are considered in [CTT11], [HS99a], [HS99b], [HS01], [LLPP15], [S94]. We define two positive weights on Γ…”

Magnetic Eigenmaps for the visualization of directed networks

“…The performance of the spectral relaxation of the cut problem can be studied using a classical result of spectral graph theory, the Cheeger inequality, which relates the Cheeger constant to the second smallest eigenvalue of the combinatorial Laplacian, providing the worst case performance for the spectral clustering method. Analogous results relate the first smallest eigenvalue of the Connection Laplacian [15] and the magnetic Laplacian [16] to a frustration quantifying the amount of inconsistency in the connection graph. In particular, the performance naturally depends on the inverse of the spectral gap 1/λ (0) 1 of the undirected measurement graph.…”

Lower bounds for the first eigenvalue of the magnetic Laplacian

“…Setting θ(r, t) 2 = ∂ ∂t , ∂ ∂t one sees that the metric takes the form (10). Finally note that θ(0, t) = 1 for all t, because F (0, ·) is the identity.…”

“…For Neumann boundary conditions, we refer in particular to the paper [9], where the authors study the multiplicity and the nodal sets corresponding to the ground state λ 1 for non-simply connected planar domains with harmonic potential (see the discussion below). Let us also mention the recent article [10] (chapter 7) where the authors establish a Cheeger type inequality for λ 1 ; that is, they find a lower bound for λ 1 (∆ A ) in terms of the geometry of Ω and the potential A. In the preprint [7], the authors approach the problem via the Bochner method.…”