Random walks on simplicial complexes and harmonics†

Abstract In this paper, we introduce a class of 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. We study an example of random walks on simplicial complexes in the context of 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 379–405, 2016

[1]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[2]  Jianbo Shi,et al.  A Random Walks View of Spectral Segmentation , 2001, AISTATS.

[3]  Caroline J. Klivans,et al.  A Cheeger-Type Inequality on Simplicial Complexes , 2012, Adv. Appl. Math..

[4]  Avi Wigderson,et al.  On the second eigenvalue of hypergraphs , 1995, Comb..

[5]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[6]  Dhruv Mubayi,et al.  Inverse Expander Mixing for Hypergraphs , 2016, Electron. J. Comb..

[7]  Ori Parzanchevski,et al.  Isoperimetric inequalities in simplicial complexes , 2012, Comb..

[8]  Andreas Krause,et al.  Advances in Neural Information Processing Systems (NIPS) , 2014 .

[9]  Kevin Françoisse,et al.  Semi-supervised Classification from Discriminative Random Walks , 2008, ECML/PKDD.

[10]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[11]  Alexander Lubotzky,et al.  Ramanujan complexes and high dimensional expanders , 2012, 1301.1028.

[12]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[13]  Matthew Kahle,et al.  Coboundary expanders , 2010, 1012.5316.

[14]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[15]  E. M. Elsayed,et al.  DIFFERENCE EQUATIONS , 2001 .

[16]  F. Chung.  Random walks and local cuts in graphs , 2007 .

[17]  Fan Chung Graham,et al.  Quasi-Random Hypergraphs , 1990, Random Struct. Algorithms.

[18]  B. Eckmann.  Harmonische Funktionen und Randwertaufgaben in einem Komplex , 1944 .

[19]  Ronald Rosenfeld,et al.  Semi-supervised learning with graphs , 2005 .

[20]  Fan Chung Graham.  Quasi-random hypergraphs revisited , 2012, Random Struct. Algorithms.

[21]  Tommi S. Jaakkola,et al.  Partially labeled classification with Markov random walks , 2001, NIPS.

[22]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[23]  Matthew Kahle,et al.  Sharp vanishing thresholds for cohomology of random flag complexes , 2012, 1207.0149.

[24]  Ori Parzanchevski,et al.  Simplicial complexes: Spectrum, homology and random walks , 2012, Random Struct. Algorithms.

[25]  Bernhard Schölkopf,et al.  Learning from Labeled and Unlabeled Data Using Random Walks , 2004, DAGM-Symposium.

[26]  Ron Rosenthal,et al.  Simplicial branching random walks and their applications , 2014, 1412.5406.

[27]  John Joseph Steenbergen,et al.  Towards a Spectral Theory for Simplicial Complexes , 2013 .