Random Walks on Simplicial Complexes for Exploring Networks
summary
Proposes a concrete chain-valued random walk on k‑simplices whose transitions use adjacent (k+1)-simplices so the generator = upward combinatorial Laplacian and the process stays in a fixed homology class. Proves ergodicity and a unique invariant measure by a norm Lyapunov estimate, simulates on a regular triangulated torus showing the invariant mass concentrates on hole boundaries, and after diffusion rescaling shows convergence of generators to a Hodge/Laplace–Beltrami limit; suggests application to faster contact tracing using cycle-valued Markov chain exploration.
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chapters (15)
- 0:00Introduction
- 1:04Epidemic modeling motivation
- 6:18Graph Laplacian recap
- 9:18Network exploration methods
- 12:38Graphs to simplicial complexes
- 18:29Simplicial vs hypergraph complexes
- 22:14Algebraic formalism chains
- 28:48Homology spaces Betti numbers
- 33:55Coboundary maps Laplacians
- 44:06Combinatorial Laplacian interpretation
- 47:05Chain-valued Markov chains
- 51:31Invariant measure Lyapunov function
- 53:43Renormalization continuum limits
- 57:04Contact tracing applications
- 1:00:06Q&A
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