Title :
Discrete Hodge Theory on Graphs: A Tutorial
Author :
Johnson, James ; Goldring, Tom
Abstract :
Hodge theory provides a unifying view of the various line, surface, and volume integrals that appear in physics and engineering applications. Hodge theory on graphs develops discrete versions of the differential forms found in the continuous theory and enables a graph decomposition into gradient, solenoidal, and harmonic components. Interpreted via similarity to gradient, curl, and Laplacian operators on vector fields, these components are useful in solving certain ranking and approximations problems that arise naturally in a graph context. This tutorial develops the rudiments of discrete Hodge theory and provides several example applications.
Keywords :
graph theory; vectors; Laplacian operators; continuous theory; curl operators; differential forms; discrete Hodge theory; gradient components; gradient operators; graph decomposition; harmonic components; solenoidal components; vector fields; Approximation algorithms; Context awareness; Discrete mathematics; Graph theory; Indexes; Scientific computing; Tutorials; Hodge decompositions; discrete Hodge theory; discrete mathematics; graph theory; scientific computing; simplicial complexes;
Journal_Title :
Computing in Science & Engineering
DOI :
10.1109/MCSE.2012.91