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 $\beta \in (0,1)$.

翻译：本文研究单源个性化PageRank（SSPPR）查询近似计算的下界问题，该查询度量从源节点$s$出发的$\alpha$衰减随机游走的概率分布。现有下界在相对误差（SSPPR-R）情形下为$\Omega\left(\min(m, 1/\delta)\right)$，在加性误差（SSPPR-A）情形下为$\Omega\left(\min(n, 1/\epsilon)\right)$，这些界仍然较为宽松。为缩小理论间隙，我们对两种设定建立了更紧的下界。针对SSPPR-R，我们证明了对于任意$\delta \in (0,1)$，存在$\Omega\left(\min\left(m, \frac{\log(1/\delta)}{\delta}\right)\right)$的下界。针对SSPPR-A，在假设图具有$m \in \mathcal{O}(n^{2-\beta})$条边（其中$\beta \in (0,1)$为任意小常数）的前提下，我们证明了对于任意$\epsilon \in (0,1)$，存在$\Omega\left(\min\left(m, \frac{\log(1/\epsilon)}{\epsilon}\right)\right)$的下界。