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2022

Combinatorial and Hodge Laplacians: Similarity and Difference

Ribando-Gros, Emily, Wang, Rui, Chen, Jiahui et al.

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As key subjects in spectral geometry and combinatorial graph theory respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures.

It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of "Hodge Laplacians on graphs" in the literature.
Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches.
To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce Boundary-Induced Graph (BIG) Laplacians using tools from Discrete Exterior Calculus (DEC).

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