Local Access to Random Walks

TL;DR

This paper develops algorithms for efficiently accessing the position of a random walk on large graphs with near-uniform distribution, providing both upper and lower bounds, and explores special cases with improved performance.

Contribution

It introduces the first sublinear-time local access algorithms for random walks on regular graphs, establishes nearly tight lower bounds, and extends results to graphs with algebraic structure and product constructions.

Findings

Efficient $ ilde{O}(rac{1}{1-eta} abla ext{sqrt}(n))$ algorithms for regular graphs.

Lower bounds of $ ilde{ ext{O}}(rac{ ext{sqrt}(n)}{ ext{log}n})$ and $ ext{O}(n^{1/4})$ for general and adversarial cases.

Polylogarithmic-time local access algorithms for abelian Cayley graphs and certain product graphs.

Abstract

For a graph $G$ on $n$ vertices, naively sampling the position of a random walk of at time $t$ requires work $\Omega(t)$. We desire local access algorithms supporting $\text{position}(G,s,t)$ queries, which return the position of a random walk from some start vertex $s$ at time $t$, where the joint distribution of returned positions is $1/\text{poly}(n)$ close to the uniform distribution over such walks in $\ell_1$ distance. We first give an algorithm for local access to walks on undirected regular graphs with $\widetilde{O}(\frac{1}{1-\lambda}\sqrt{n})$ runtime per query, where $\lambda$ is the second-largest eigenvalue in absolute value. Since random $d$-regular graphs are expanders with high probability, this gives an $\widetilde{O}(\sqrt{n})$ algorithm for $G(n,d)$, which improves on the naive method for small numbers of queries. We then prove that no that algorithm with sub-constant error given probe access to random $d$-regular graphs can have runtime better than $\Omega(\sqrt{n}/\log(n))$ per query in expectation, obtaining a nearly matching lower bound. We further show an $\Omega(n^{1/4})$ runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime $\text{polylog}(n)$ per query. This also allows for efficient local access to walks on $\text{polylog}$ degree expanders. We extend our results to graphs constructed using the tensor product (giving local access to walks on degree $n^\epsilon$ graphs for any $\epsilon \in (0,1]$) and Cartesian product.