Journal
SIAM REVIEW
Volume 62, Issue 3, Pages 685-715Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1223101
Keywords
cohomology; Hodge decomposition; Hodge Laplacians; graphs
Categories
Funding
- AFOSR [FA9550-13-1-0133]
- DARPA [D15AP00109]
- NSF [IIS 1546413, DMS 1209136, DMS 1057064]
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This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is simplicity, requiring only knowledge of linear algebra and graph theory. We have also isolated the algebra from the topology to show that a large part of cohornology and Hodge theory is nothing more than the linear algebra of matrices satisfying AB = 0. For the remaining topological aspect, we cast our discussion entirely in terms of graphs as opposed to less familiar topological objects like simplicial complexes.
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