We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector $\pi \in \mathbb{R}^n$ for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive ($L_1$) approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case $(1/\alpha)^{\Theta(\log \log n)}$ update time is necessary and sufficient for this problem, where $\alpha$ is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires $\Omega(n^{1-\delta})$ time per update on average, for any $\delta > 0$, even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time $\Omega(n^{1-\delta})$, for any $\delta > 0$, even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al.~(VLDB 2010). This sharply contrasts with the undirected setting, where we show that $\rm{poly}\ \log n$ update time is feasible, even in the fully dynamic setting under oblivious adversary.

翻译：我们考虑动态环境下的PageRank问题，目标是在边插入和删除序列的图中明确维护一个近似PageRank向量$\pi \in \mathbb{R}^n$。我们的主要结果是对动态PageRank维护在乘性和加性（$L_1$）近似下的复杂性进行了完整刻画。首先，我们在增量和减量环境下为维护加性近似PageRank建立了匹配的下界和上界。特别地，我们证明在最坏情况下，该问题需要$(1/\alpha)^{\Theta(\log \log n)}$的更新时间是必要且充分的，其中$\alpha$是所需的加性近似。另一方面，我们证明常用的ForwardPush方法的运行时间远差于这一最优时间。具体而言，我们表明即使在增量环境下，ForwardPush平均每次更新需要$\Omega(n^{1-\delta})$时间，对于任意$\delta > 0$。然而，对于乘性近似，我们证明情况显著更具挑战性。具体地，我们证明任何明确维护有向图PageRank向量常数因子乘性近似的算法，即使在增量环境下，必须具有摊销更新时间$\Omega(n^{1-\delta})$，对于任意$\delta > 0$，从而解决了Bahmani等人（VLDB 2010）提出的一个13年之久的开放问题。这与无向图设置形成鲜明对比，我们证明在完全动态环境下，即使面对不知情对手，$\rm{poly}\ \log n$的更新时间也是可行的。