We study Personalized PageRank (PPR), where for nodes $s,t$ in a graph $G$, $\pi(s,t)$ is the probability that an $\alpha$-decay random walk from $s$ ends at $t$. Two key queries are: Single-Source PPR (SSPPR), computing $\pi(s,\cdot)$ for fixed $s$, and Single-Target PPR (STPPR), computing $\pi(\cdot,t)$ for fixed $t$. SSPPR is studied under absolute error (SSPPR-A), requiring $|\hat{\pi}(s,t)-\pi(s,t)|\le \epsilon$, and relative error (SSPPR-R), requiring $|\hat{\pi}(s,t)-\pi(s,t)|\le c\pi(s,t)$ for $t$ with $\pi(s,t)\ge \delta$; STPPR adopts the same relative criterion. These queries support web search, recommendation, sparsification, and graph neural networks. The best known upper bounds are $O(\min(\tfrac{\log(1/\epsilon)}{\epsilon^{2}},\tfrac{\sqrt{m\log n}}{\epsilon},m\log\tfrac{1}{\epsilon}))$ for SSPPR-A and $O(\min(\tfrac{\log(1/\delta)}{\delta},\sqrt{\tfrac{m\log n}{\delta}},m\log\tfrac{\log n}{\delta m}))$ for SSPPR-R, while lower bounds remain $\Omega(\min(n,1/\epsilon))$, $\Omega(\min(m,1/\delta))$, and $\Omega(\min(n,1/\delta))$, leaving large gaps. We close these gaps by (i) presenting a Monte Carlo algorithm that tightens the SSPPR-A upper bound to $O(1/\epsilon^{2})$, and (ii) proving, via an arc-centric construction, lower bounds $\Omega(\min(m,\tfrac{\log(1/\delta)}{\delta}))$ for SSPPR-R, $\Omega(\min(m,\tfrac{1}{\epsilon^{2}}))$ (and intermediate $\Omega(\min(m,\tfrac{\log(1/\epsilon)}{\epsilon}))$) for SSPPR-A, and $\Omega(\min(m,\tfrac{n}{\delta}\log n))$ for STPPR. For practical settings ($\delta=\Theta(1/n)$, $\epsilon=\Theta(n^{-1/2})$, $m\in\Omega(n\log n)$) these bounds meet the best known upper bounds, establishing the optimality of Monte Carlo and FORA for SSPPR-R, our algorithm for SSPPR-A, and RBS for STPPR, and yielding a near-complete complexity landscape for PPR queries.

翻译：我们研究个性化PageRank（PPR），其中对于图$G$中的节点$s,t$，$\pi(s,t)$表示从$s$出发的$\alpha$-衰减随机游走终止于$t$的概率。两个关键查询是：单源PPR（SSPPR），计算固定$s$对应的$\pi(s,\cdot)$；以及单目标PPR（STPPR），计算固定$t$对应的$\pi(\cdot,t)$。SSPPR在绝对误差（SSPPR-A）条件下被研究，要求$|\hat{\pi}(s,t)-\pi(s,t)|\le \epsilon$；在相对误差（SSPPR-R）条件下被研究，要求对于满足$\pi(s,t)\ge \delta$的$t$，有$|\hat{\pi}(s,t)-\pi(s,t)|\le c\pi(s,t)$；STPPR采用相同的相对误差标准。这些查询支持网络搜索、推荐系统、图稀疏化以及图神经网络。已知最佳上界为：SSPPR-A的$O(\min(\tfrac{\log(1/\epsilon)}{\epsilon^{2}},\tfrac{\sqrt{m\log n}}{\epsilon},m\log\tfrac{1}{\epsilon}))$，SSPPR-R的$O(\min(\tfrac{\log(1/\delta)}{\delta},\sqrt{\tfrac{m\log n}{\delta}},m\log\tfrac{\log n}{\delta m}))$，而下界仍为$\Omega(\min(n,1/\epsilon))$、$\Omega(\min(m,1/\delta))$和$\Omega(\min(n,1/\delta))$，存在较大差距。我们通过以下方式弥合了这些差距：（i）提出一种蒙特卡洛算法，将SSPPR-A的上界收紧至$O(1/\epsilon^{2})$；（ii）通过基于弧的构造，证明SSPPR-R的下界为$\Omega(\min(m,\tfrac{\log(1/\delta)}{\delta}))$，SSPPR-A的下界为$\Omega(\min(m,\tfrac{1}{\epsilon^{2}}))$（以及中间的$\Omega(\min(m,\tfrac{\log(1/\epsilon)}{\epsilon}))$），STPPR的下界为$\Omega(\min(m,\tfrac{n}{\delta}\log n))$。在实际设置（$\delta=\Theta(1/n)$，$\epsilon=\Theta(n^{-1/2})$，$m\in\Omega(n\log n)$）下，这些下界与已知最佳上界相匹配，从而确立了蒙特卡洛方法和FORA对于SSPPR-R、我们的算法对于SSPPR-A、以及RBS对于STPPR的最优性，并为PPR查询构建了近乎完整的计算复杂度图景。