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，目标是在一系列边插入和删除操作中显式维护图的近似PageRank向量$\pi \in \mathbb{R}^n$。我们的主要成果是对乘法和加法（$L_1$）近似两种情况下动态PageRank维护复杂度的完整刻画。首先，我们在增量式和减量式两种设定下建立了维护加法近似PageRank的匹配下界与上界。具体而言，我们证明对于该问题，最坏情况下$(1/\alpha)^{\Theta(\log \log n)}$的更新时间是必要且充分的，其中$\alpha$为目标加法近似精度。另一方面，我们证明常用的ForwardPush方法在此最优运行时间下表现显著更差。具体地，我们指出即使仅在增量式设定下，ForwardPush每次更新平均需要$\Omega(n^{1-\delta})$时间，其中$\delta > 0$为任意常数。然而对于乘法近似，我们展示情况更为复杂。具体地，我们证明任何显式维护有向图PageRank向量常数因子乘法近似的算法，其均摊更新时间必须为$\Omega(n^{1-\delta})$（$\delta > 0$为任意常数），即使在增量式设定下也是如此，从而解决了Bahmani等人（VLDB 2010）一个13年之久的开放问题。这与无向图设定形成鲜明对比：在对抗性攻击下的完全动态设定中，无向图可以实现$\rm{poly}\ \log n$的更新时间。