Estimating Random-Walk Probabilities in Directed Graphs

Abstract

We study discounted random walks in directed graphs. In each step, the walk either terminates with a constant probability $\alpha$, or proceeds to a random out-neighbor. Our goal is to estimate the probability $\pi(s, t)$ that a discounted random walk starting from $s$ terminates at $t$. This probability is also known as the Personalized PageRank (PPR) score, which measures the relevance of $t$ to $s$, for instance, when $s$ and $t$ are web pages on the Internet. We aim to estimate $\pi(s, t)$ within a constant relative error with constant probability. A variety of algorithms have been developed for several problem variants, such as single-pair, single-source, single-target, and single-node estimation, under both worst-case and average-case settings, and for different combinations of allowed graph queries. However, in many important cases, there remain polynomial gaps between known upper and lower bounds. In this paper, we establish tight upper and lower bounds (up to logarithmic factors of $n$) for all problem variants and query combinations, closing all existing gaps in both the worst-case and average-case settings. Below we give some examples for the worst-case settings. As an upper-bound example, the classic power method estimates $\pi(s,t)$ if it is above a threshold $\delta$ in time $O(m\log(1/\delta))$ but $\pi(s,t)$ can be as small as $1/n^{\Theta(n)}$. For contrast, we propose algorithms that deterministically estimate arbitrarily small $\pi(s,t)$ in $O(m\log n)$ time. As a lower-bound example, we improve the lower bound for the single-pair problem from $\Omega(\min\{n,1/\delta\})$ to $\Omega(\min\{m,1/\delta\})$, which is optimal (up to logarithmic factors) since a simple Monte Carlo estimate takes $O(1/\delta)$ time.