Combinatorial and Hodge Laplacians: Similarities and Differences

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. For example, the spectral decomposition of a vector field on a simple point cloud using combinatorial Laplacians defined on some commonly used simplicial complexes does not give rise to the same curl-free and divergence-free components that one would obtain from the spectral decomposition of a vector field using either the continuous Hodge Laplacians defined on differential forms in manifolds or the discretized Hodge Laplacians defined on a point cloud with boundary in the Eulerian representation or on a regular mesh in the Eulerian representation. 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). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences among the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as level-set functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.