Near-Optimality for Single-Source Personalized PageRank

TL;DR

This paper advances the theoretical understanding of Single-Source Personalized PageRank (SSPPR) by establishing near-optimal bounds, closing the gap between existing upper and lower bounds, and extending results to related PageRank queries.

Contribution

It improves upper bounds for SSPPR queries, establishes stronger lower bounds, and demonstrates near-optimality, including for the first time for SSPPR and related Single-Target PageRank.

Findings

Upper bounds for SSPPR-A and SSPPR-R improved to near-optimal levels.

Lower bounds for SSPPR-A and SSPPR-R strengthened, matching upper bounds in many regimes.

Lower bound for Single-Target Personalized PageRank (STPPR) improved, matching its upper bound.

Abstract

The \emph{Single-Source Personalized PageRank} (SSPPR) query is central to graph OLAP, measuring the probability $\pi(s,t)$ that an $\alpha$-decay random walk from node $s$ terminates at $t$. Despite decades of research, a significant gap remains between upper and lower bounds for its computational complexity. Existing upper bounds are $O\left(\min\left(\frac{\log(1/\epsilon)}{\epsilon^2}, \frac{\sqrt{m \log n}}{\epsilon}, m \log \frac{1}{\epsilon}\right)\right)$ for SSPPR-A and $O\left(\min\left(\frac{\log(1/n)}{\delta}, \sqrt{m \log(n/\delta)}, m \log \left(\frac{\log(n)}{m\delta}\right)\right)\right)$ for SSPPR-R, with trivial lower bounds of $\Omega(\min(n,1/\epsilon))$ and $\Omega(\min(n,1/\delta))$. This work narrows or closes this gap. We improve the upper bounds for SSPPR-A and SSPPR-R to $O\left(\frac{1}{\epsilon^2}\right)$ and $O\left(\min\left(\frac{\log(1/\delta)}{\delta}, m + n \log(n) \log \left(\frac{\log(n)}{m\delta}\right)\right)\right)$, respectively, offering improvements by factors of $\log(1/\epsilon)$ and $\log\left(\frac{\log(n)}{m\delta}\right)$. On the lower bound side, we establish stronger results: $\Omega(\min(m, 1/\epsilon^2))$ for SSPPR-A and $\Omega(\min(m, \frac{\log(1/\delta)}{\delta}))$ for SSPPR-R, strengthening theoretical foundations. Our upper and lower bounds for SSPPR-R coincide for graphs with $m \in \Omega(n \log^2 n)$ and any threshold $\delta, 1/\delta \in O(\text{poly}(n))$, achieving theoretical optimality in most graph regimes. The SSPPR-A query attains partial optimality for large error thresholds, matching our new lower bound. This is the first optimal result for SSPPR queries. Our techniques generalize to the Single-Target Personalized PageRank (STPPR) query, improving its lower bound from $\Omega(\min(n, 1/\delta))$ to $\Omega(\min(m, \frac{n}{\delta} \log n))$, matching the upper bound and revealing its optimality.