PageRank Centrality in Directed Graphs with Bounded In-Degree

TL;DR

This paper investigates the complexity of estimating a node's PageRank centrality in directed graphs with bounded in-degree, establishing tight bounds and proposing a new algorithm with a novel randomized backward propagation technique.

Contribution

The paper provides a tight lower bound for PageRank estimation complexity and introduces a new algorithm that matches this bound up to logarithmic factors.

Findings

Established a tight lower bound for PageRank estimation complexity.

Proposed a new algorithm with a randomized backward propagation technique.

Identified a gap in complexity bounds related to in-degree bounds.

Abstract

We study the computational complexity of locally estimating a node's PageRank centrality in a directed graph $G$. For any node $t$, its PageRank centrality $\pi(t)$ is defined as the probability that a random walk in $G$, starting from a uniformly chosen node, terminates at $t$, where each step terminates with a constant probability $\alpha\in(0,1)$. To obtain a multiplicative $\big(1\pm O(1)\big)$-approximation of $\pi(t)$ with probability $\Omega(1)$, the previously best upper bound is $O(n^{1/2}\min\{ \Delta_{in}^{1/2},\Delta_{out}^{1/2},m^{1/4}\})$ from [Wang, Wei, Wen, Yang, STOC '24], where $n$ and $m$ denote the number of nodes and edges in $G$, and $\Delta_{in}$ and $\Delta_{out}$ upper bound the in-degrees and out-degrees of $G$, respectively. Using a refinement of the proof in the same paper, we establish a lower bound of $\Omega(n^{1/2}\min\{\Delta_{in}^{1/2}/n^{\gamma},\Delta_{out}^{1/2}/n^{\gamma},m^{1/4}\})$, where $\gamma=\frac{1}{2}(2\max\{\log_{1/(1-\alpha)}\Delta_{in},1\}-1)^{-1}$. As $\gamma$ only depends on $\Delta_{in}$ and $n^{\gamma}=O(1)$ for $\Delta_{in}=\Omega\left(n^{\Omega(1)}\right)$, the known upper bound is tight if we only parameterize the complexity by $n$, $m$, and $\Delta_{out}$. However, there remains a gap of $\Omega(n^{\gamma})$ when considering $\Delta_{in}$, and this gap is large when $\Delta_{in}$ is small. In the extreme case where $\Delta_{in}\le1/(1-\alpha)$, we have $\gamma=1/2$, leading to a gap of $\Omega(n^{1/2})$ between the bounds $O(n^{1/2})$ and $\Omega(1)$. In this paper, we present a new algorithm that achieves the above lower bound (up to logarithmic factors). The algorithm assumes that $n$ and the bounds $\Delta_{in}$ and $\Delta_{out}$ are known in advance. Our key technique is a novel randomized backwards propagation process that only propagates selectively based on Monte Carlo estimated PageRank scores.