Spectra of combinatorial Laplace operators on simplicial complexes

By Danijela Horak, Jürgen Jost. 2011

297 citations recorded by the source corpus.

Abstract We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator Δ i u p on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1818625085a40fca5895927d7b0d50bfb2639c8b

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Networks beyond pairwise interactions: structure and dynamics (2020, background): “It is in fact straight-forward to define Laplacian operators for any dimension on simplicial complexes [67, 68], they can approximate both regular manifolds and highly irregular structures [69, 70], and they come naturally equipped with boundary operators stringing together simplices with different ”
Random Walks on Simplicial Complexes and the normalized Hodge Laplacian (2018, background): “The normalized Hodge Laplacian we consider here is of a ``standard form"" [62, 68] and admits a (normalized) Hodge decomposition, which is of interest for our applications.”
Weighted persistent homology (2017, unknown): “[9, 16, 23]), and our definition is not unique.”
What are higher-order networks? (2021, methodology): “An interesting alternative way to use the boundary maps ∂n to extract information about a simplicial complex is by constructing Hodge-Laplacians [94, 126,138,148,187]”