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

TL;DR

This paper extends the concept of random walks and spectral analysis from graphs to simplicial complexes by normalizing the Hodge Laplacian, enabling new insights into higher-order data interactions.

Contribution

It introduces a normalized Hodge Laplacian for simplicial complexes and links it to random walks on edges, advancing Laplacian-based analytics for complex topological data.

Findings

Derived spectral embeddings for trajectory data analysis.

Developed a generalized personalized PageRank for edge-space.

Demonstrated applications on ocean drifter trajectories and book co-purchasing data.

Abstract

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.