Combinatorics

Discrete mathematics

Graph

Abstract simplicial complex

Graph power

Simplicial set

Pure mathematics

Mathematics

Random walk

Statistics

Laplacian matrix

Laplace operator

Mathematical analysis

Simplicial complex

Line graph

Resistance distance

Homotopy category

Homotopy

Simplicial manifold

SIAM Review

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.

Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian