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Random Walks on Simplicial Complexes and Harmonics
Combinatorics · Submitted 18 Oct 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $1 \leq k \leq d$ and which relates to the local random walk on a graph defined by Fan Chung.
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The abstract, sentence by sentence
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 $1 \leq k \leq d$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we introduce random walks with absorbing states on simplicial complexes. | contribution 1.00 | 0.92 | 0.09 | asserted 0.99 |
| s1 | 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 $1 \leq k \leq d$ and which relates to the local random walk on a graph defined by Fan Chung. | method 0.61 | 0.78 | 0.16 | asserted 1.00 |
| s2 | We also examine an application of random walks on simplicial complexes to a semi-supervised learning problem. | contribution 1.00 | 0.91 | 0.10 | asserted 0.97 |
| s3 | Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. | method 0.54 | 0.87 | 0.08 | asserted 0.99 |
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The paper
Original 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 $1 \leq k \leq d$ 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.
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How the literature uses it
4 arXiv papers cite this in 10 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
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In this work, we use the gradient estimation method by [ref] which aims to solve the (local) weighted linear regression on the local tangent plane.
Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data · Velocity field mapping between representations · 0.99
Compares with it 1 paper
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In a subsequent work [ref] , Mukherjee and Steenbergen constructed a similar model for random walks on simplicial complexes, which is connected to the lower Laplacian and in particular allows to study the top homology of the complex.
Simplicial branching random walks and their applications · Introduction · 0.81
Cites it as background 4 papers
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A corresponding neighboring relation using faces instead of cofaces was defined in [ref] .
Simplicial branching random walks and their applications · Simplicial complexes · 0.92
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On the theoretical front, frameworks of random walks on simplices of down [ref] and up [ref] $k$-Laplacian have also been visited.
Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data · Related works and discussion · 0.96
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In addition, the adjacency matrix can be interpreted as a generator of a stochastic process which is a high-dimensional analog of simple random walks, see [ref] for more details.
Eigenvalue confinement and spectral gap for random simplicial complexes · Introduction · 0.92
What it cites · 9 sentences without a confident label
Compares with 1 paper
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One is that the norm of the normalized marginal difference for the Neumann $k$-walk starting at a single oriented simplex stays bounded away from 0 (see Proposition 2.8 of [ref] ), whereas this need not hold for the Dirichlet $k$-walk (as in Corollary [ref]).
Simplicial complexes: spectrum, homology and random walks · Random walks with Neumann boundary conditions · 0.99
Cites as background 1 paper
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Disorientability was defined in [ref] and shown to be a higher-dimensional analogue of bipartiteness for graphs.
Simplicial complexes: spectrum, homology and random walks · The Laplacian · 0.88
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