Simulating Random Walks in Random Streams

TL;DR

This paper introduces a space-efficient single-pass streaming algorithm for simulating nearly independent random walks in random streams, enabling improved graph analysis tasks like PageRank estimation.

Contribution

It presents the first space-efficient algorithm for simulating random walks in random streams, solving an open problem and broadening graph streaming applications.

Findings

Efficient approximation of k-step walk distributions with space complexity depending on error and walk length.

Application to estimating return probabilities and PageRank in streaming models.

Impossibility results for directed graphs highlight limitations of the approach.

Abstract

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 $\varepsilon$ per walk using $(1/\varepsilon)^{O(k)} 2^{O(k^2)}\cdot b$ 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 $k^\text{th}$ 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.