Random Walks on Simplicial Complexes and Harmonics

2013/10/18 by Mukherjee, Sayan, Steenbergen, John · 3 citations
#05C81 #35P05 #55N10 #55U10 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1310.5099

Abstract

In this paper, we introduce random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k-dimensional Laplacian for 1 ≤ k ≤ d and which relates to the local random walk on a graph defined by Fan Chung. We also examine an application of random walks on simplicial complexes to a semi-supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs.

Cited by

• New class of exactly flat topological bands - compact localised states protected by local graph topology
• From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions
• Hall's theorem for reconfigurations and higher dimensional topological connectedness

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