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Issue No.05 - Sept.-Oct. (2013 vol.15)
pp: 42-55
James L. Johnson , Western Washington University
Tom Goldring , US National Security Agency
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.
INDEX TERMS
Graph theory, Context awareness, Indexes, Tutorials, Approximation algorithms, Scientific computing, Discrete mathematics,scientific computing, discrete Hodge theory, graph theory, discrete mathematics, simplicial complexes, Hodge decompositions
CITATION
James L. Johnson, Tom Goldring, "Discrete Hodge Theory on Graphs: A Tutorial", Computing in Science & Engineering, vol.15, no. 5, pp. 42-55, Sept.-Oct. 2013, doi:10.1109/MCSE.2012.91
REFERENCES
1. D.N. Arnold,R.S. Falk,, and R. Winther,“Finite Element Exterior Calculus: from Hodge Theory to Numerical Stability,” Bulletin Am. Mathematical Soc., vol. 47, no. 2, 2010, pp. 281-354.
2. X. Jiang et al., “Statistical Ranking and Combinatorial Hodge Theory,” Mathematical Programming, vol. 127, no. 1, 2011, pp. 203-244.
3. X. Jiang et al., “Learning to Rank with Combinatorial Hodge Theory,” research paper, Dept. Mathematics, Stanford Univ., 2008; http://comptop.stanford.edu/u/preprintshodge-preprint2.pdf .
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