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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 ‐partite simplicial complex, which we callexpanderized higher‐order random walks—where the sequence of updated coordinates corresponds to the sequence of vertices visited by a random walk over an auxiliary expander graph . When is the clique with self‐loops on , this random walk reduces to the usual down‐up walk, and when is the directed cycle on , 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 auxiliary graph . Our construction can be thought of 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) for sampling uniformly random list colorings of a graph of maximum degree where each vertex has at least and at most colors, (ii) for sampling the Ising model with a PSD interaction matrix satisfying and the external field —here the notation hides a constant that depends linearly on the largest entry of . 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 ‐entropy contraction in simplicial complexes—giving simpler proofs for many pre‐existing results.
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Simulating Random Walks in Random Streams
The random order graph streaming model has received significant attention recently, with problems such as matching size estimation, component counting, and the evaluation of bounded degree constant query testable properties shown to admit surprisingly space efficient algorithms. The main result of this paper is a space efficient single pass random order streaming algorithm for simulating nearly independent random walks that start at uniformly random vertices. We show that the distribution of k-step walks from b vertices chosen uniformly at random can be approximated up to error ∊ per walk using  words of space with a single pass over a randomly ordered stream of edges, solving an open problem of Peng and Sohler [SODA '18]. Applications of our result include the estimation of the average return probability of the k-step walk (the trace of the kth power of the random walk matrix) as well as the estimation of PageRank. We complement our algorithm with a strong impossibility result for directed graphs.
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- Award ID(s):
- 1751040
- PAR ID:
- 10336438
- Date Published:
- Journal Name:
- Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
- ISSN:
- 1071-9040
- Page Range / eLocation ID:
- 3091-3126
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1137/1.9781611977073.120
