Education

SIAM Review, Volume 66, Issue 3, Page 573-573, May 2024.
In this issue the Education section presents “Combinatorial and Hodge Laplacians: Similarities and Differences,” by Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, and Guo-Wei Wei. Combinatorial Laplacians and their spectra are important tools in the study of molecular stability, electrical networks, neuroscience, deep learning, signal processing, etc. The continuous Hodge Laplacian allows one, in some cases, to generate an unknown shape from only its Laplacian spectrum. In particular, both combinatorial Laplacians and continuous Hodge Laplacian are useful in describing the topology of data; see, for instance, [L.-H. Lim, “Hodge Laplacians on graphs,” SIAM Rev., 62 (2020), pp. 685--715]. Since nowadays computations frequently involve these Laplacians, it is important to have a good understanding of the differences and relations between them. Indeed, though the Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data, at the same time they are intrinsically different in their domains of definitions and applicability to specific data formats. To facilitate comparisons, the authors introduce boundary-induced graph (BIG) Laplacians, the purpose of which is “to put the combinatorial Laplacians and Hodge Laplacian on equal footing.” BIG Laplacian brings, in fact, the combinatorial Laplacian closer to the continuous Hodge Laplacian. In this paper similarities and differences between combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are examined. Some elements of spectral analysis related to topological data analysis (TDA) are also provided. TDA and connected ideas have recently gained a lot of interest, and so this paper is timely. It is written in a way that should make it accessible for early career researchers; the reader should already have a good understanding of some notions of graph theory, spectral geometry, differential geometry, and algebraic topology. The paper is not self-contained and eventually could be used by group-based research projects in a Master's program for advanced mathematics students.