Random Walks on Simplicial Complexes and the normalized Hodge Laplacian

By Michael T. Schaub, Austin R. Benson, Paul Horn, Gábor Lippner, Ali Jadbabaie. SIAM Review, 2018

339 citations recorded by the source corpus.

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 re...

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ac594bbf3dcc8767acf7358aa4f0768701a3553b

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Citations in the corpus

Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks (2021, methodology): “The k-th Hodge Laplacian of the simplicial complex (Lim, 2015; Barbarossa & Sardellitti, 2020; Schaub et al., 2020), a diffusion operator for signals defined over the oriented k-simplices is defined as Lk = B > k Bk +Bk+1B > k+1.”
Dynamics on higher-order networks: a review (2022, background): “[121], who propose a normalized Hodge Laplacian matrix and demonstrate how it is associated with random walk dynamics on simplicial complexes, specifically on edges.”
Weisfeiler and Lehman Go Cellular: CW Networks (2021, methodology): “This is very useful because finding normalised versions of Hodge Laplacians is not trivial and even on simplicial complexes [63], the process of constructing one can be quite involved.”
What are higher-order networks? (2021, methodology): “The three subspaces of the Hodge-decomposition are generally spanned by certain subsets of eigenvectors of Ln [16,234,237].”
Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching (2019, background): “or analysis and computation in high dimensional data sets, e.g., using persistent homologies [15], little is understood about their role in shaping dynamical processes, save for a handful of examples [21, 22]. Using brain dynamics as a motivating application, we study the dynamics of heterogeneous p”
Hodge Laplacians on graphs (2015, background): “” Unlike physical problems arising from areas such as continuum mechanics or electromagnetics, where the differentiable Hodge–de Rham theory has been applied with great efficacy for both modeling and computations [1, 2, 27, 40, 47, 58, 59], those arising from data analytic applications are likely to”