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Expanderizing Higher Order Random Walks

Authors Vedat Levi Alev , Shravas Rao

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ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Generating random combinatorial structures
Keywords
  • Higher Order Random Walks
  • Expander Graphs
  • Glauber Dynamics
  • Derandomized Squaring
  • High Dimensional Expansion
  • Spectral Independence
  • Entropic Independence

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Abstract

We study a variant of the down-up (also known as the Glauber dynamics) and up-down walks over an n-partite simplicial complex, which we call expanderized higher order random walks - where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph H. When H is the clique with self loops on [n], this random walk reduces to the usual down-up walk and when H is the directed cycle on [n], this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph H. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan (RANDOM 2005).
We study the mixing times of our expanderized walks in two example cases: We show that when initiated with an expander graph our expanderized random walks have mixing time (i) O(n log n) for sampling a uniformly random list colorings of a graph G of maximum degree Δ = O(1) where each vertex has at least (11/6 - ε) Δ and at most O(Δ) colors, (ii) O_h((n log n)/(1 - ‖J‖_op)²) for sampling the Ising model with a PSD interaction matrix J ∈ ℝ^{n×n} satisfying ‖J‖_op ≤ 1 and the external field h ∈ ℝⁿ- here the O(•) notation hides a constant that depends linearly on the largest entry of h. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor.
We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global Φ-entropy contraction in simplicial complexes - giving simpler proofs for many pre-existing results.

Cite As Get BibTex

Vedat Levi Alev and Shravas Rao. Expanderizing Higher Order Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 58:1-58:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.58

Author Details

Vedat Levi Alev
  • Hebrew University of Jerusalem, Israel
Shravas Rao
  • Portland State University, Portland, OR, United States of America

Funding

  • Alev, Vedat Levi: Supported by the ERC grant of Alex Lubotzky (European Union’s Horizon 2020/882751), the ISF grant 2669/21 and ERC grant 834735 of Gil Kalai, and ISF grant 2990/21 of Ori Parzanchevski.
  • Rao, Shravas: This material is based upon work supported by the National Science Foundation under Award No. 2348489.

Acknowledgements

We would like to thank Fernando Granha Jeronimo for many insightful discussions concerning expander graphs and anonymous referees for their many insightful comments.

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