Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian
Abstract
Using graphs to model pairwise relationships between entities is a ubiquitous framework for studying complex systems and data. Simplicial complexes extend this dyadic model of graphs to polyadic relationships and have emerged as a model for multinode relationships occurring in many complex systems. For instance, biological interactions occur between sets of molecules and communication systems include group messages that are not pairwise interactions. While Laplacian dynamics have been intensely studied for graphs, corresponding notions of Laplacian dynamics beyond the node-space have so far remained largely unexplored for simplicial complexes. In particular, diffusion processes such as random walks and their relationship to the graph Laplacian---which underpin many methods of network analysis, including centrality measures, community detection, and contagion models---lack a proper correspondence for general simplicial complexes.
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 toward 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 copurchasing dataset.
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Submitted: 17 July 2018
Accepted: 19 August 2019
Published online: 7 May 2020
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Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-13-1-0097
Army Research Office https://doi.org/10.13039/100000183 : W911NF-19-1-0057
Office of Defense Programs https://doi.org/10.13039/100006127
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 702410
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1830274
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