2018/07/31 by Michael T. Schaub, Austin R. Benson, Paul Horn +2 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Abstract simplicial complex #Complex Network Analysis Techniques #Data Visualization and Analytics #Generalization #Graph #Laplace operator #Laplacian matrix #PageRank #Random walk #Simplicial complex #Spectral graph theory #Topological and Geometric Data Analysis #cs.DM #cs.SI #math.AT #physics.soc-ph
paper · pdf · doi:10.1137/18m1201019
published as SIAM Review 2020 62:2, 353-391 · 38 pages, 11 figures, 1 table (abstract above shortened); to appear in SIAM Review, June 2020
openalex created_date 2018/08/03 · arxiv created 2019/11/06 · openalex publication_date 2020/01/01 · arxiv updated 2020/05/08 · openalex updated_date 2026/08/06
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.