Hodge Laplacians on graphs

By Lek-Heng Lim. 2015

312 citations recorded by the source corpus.

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 cohomology and Hodge theory is nothing more than the linear algebra of matrices satisfying $AB = 0$. For the remaining topological aspect, we cast our discussions entirely in terms of graphs as opposed to less-familiar topological objects like simplicial complexes.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Differentiable Physics-informed Graph Networks (2019, background): “Since f and F can be viewed as scalar ﬁelds and tangent vector ﬁelds in differential geometry (Figure 2), the corresponding discrete operators on graphs can be deﬁned as follow (Lim, 2015).”
Physics-aware Difference Graph Networks for Sparsely-Observed Dynamics (2020, background): “While the difference operators are generalized in Riemannian manifolds (Lai et al., 2013; Lim, 2015), there exist numerical error compared to those in continuous space and the error can be larger when the nodes are spatially far from neighboring nodes because the connected nodes ( j ∈ N i ) of i…”
Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks (2021, background): “The k-th Hodge Laplacian of the simplicial complex (Lim, 2015; Barbarossa & Sardellitti, 2020; Schaub et al., 2020), a diffusion operator for signals defined over the oriented k-simplices is defined as Lk = B > k Bk +Bk+1B > k+1.”
Random Walks on Simplicial Complexes and the normalized Hodge Laplacian (2018, methodology): “Indeed, the operator B2 is a discrete analog of the familiar notion of a curl in vector calculus [85].”
What are higher-order networks? (2021, methodology): “To understand why this is the case, it is insightful to introduce the Hodgedecomposition [126, 187, 251].”