Pseudorandomness of the Sticky Random Walk

TL;DR

This paper investigates the pseudorandomness properties of a generalized sticky random walk on expander graphs, providing tighter bounds on total variation distance and establishing connections between the walk and expander graph families.

Contribution

It proves a conjecture about TVD bounds for a family of expanders and introduces a generalized sticky random walk with improved pseudorandomness guarantees.

Findings

TVD bound improved to O(λ) for certain expanders

Generalized sticky random walk parameterizes an infinite family of expanders

Fourier and combinatorial methods used to analyze pseudorandomness

Abstract

We extend the pseudorandomness of random walks on expander graphs using the sticky random walk. Building on prior works, it was recently shown that expander random walks can fool all symmetric functions in total variation distance (TVD) upto an $O(\lambda(\frac{p}{\min f})^{O(p)})$ error, where $\lambda$ is the second largest eigenvalue of the expander, $p$ is the size of the arbitrary alphabet used to label the vertices, and $\min f = \min_{b\in[p]} f_b$, where $f_b$ is the fraction of vertices labeled $b$ in the graph. Golowich and Vadhan conjecture that the dependency on the $(\frac{p}{\min f})^{O(p)}$ term is not tight. In this paper, we resolve the conjecture in the affirmative for a family of expanders. We present a generalization of the sticky random walk for which Golowich and Vadhan predict a TVD upper bound of $O(\lambda p^{O(p)})$ using a Fourier-analytic approach. For this family of graphs, we use a combinatorial approach involving the Krawtchouk functions to derive a strengthened TVD of $O(\lambda)$. Furthermore, we present equivalencies between the generalized sticky random walk, and, using linear-algebraic techniques, show that the generalized sticky random walk parameterizes an infinite family of expander graphs.