Tighter Lower Bounds for Single Source Personalized PageRank

Abstract: We study lower bounds for approximating the Single Source Personalized PageRank (SSPPR) query, which measures the probability distribution of an $\alpha$-decay random walk starting from a source node $s$. Existing lower bounds remain loose-$\Omega\left(\min(m, 1/\delta)\right)$ for relative error (SSPPR-R) and $\Omega\left(\min(n, 1/\epsilon)\right)$ for additive error (SSPPR-A). To close this gap, we establish tighter bounds for both settings. For SSPPR-R, we show a lower bound of $\Omega\left(\min\left(m, \frac{\log(1/\delta)}{\delta}\right)\right)$ for any $\delta \in (0,1)$. For SSPPR-A, we prove a lower bound of $\Omega\left(\min\left(m, \frac{\log(1/\epsilon)}{\epsilon}\right)\right)$ for any $\epsilon \in (0,1)$, assuming the graph has $m \in \mathcal{O}(n^{2-\beta})$ edges for any arbitrarily small constant $\beta \in (0,1)$.