Revisiting Local Computation of PageRank: Simple and Optimal

TL;DR

The computational complexity of locally estimating a node’s PageRank centrality is investigated and the matching upper and lower bounds resolve the open problem of whether one can tighten the bounds given by Bressan, Peserico, and Pretto.

Abstract

We revisit the classic local graph exploration algorithm ApproxContributions proposed by Andersen, Borgs, Chayes, Hopcroft, Mirrokni, and Teng (WAW '07, Internet Math. '08) for computing an $ε$-approximation of the PageRank contribution vector for a target node $t$ on a graph with $n$ nodes and $m$ edges. We give a worst-case complexity bound of ApproxContributions as $O(nπ(t)/ε\cdot\min(Δ_{in},Δ_{out},\sqrt{m}))$, where $π(t)$ is the PageRank score of $t$, and $Δ_{in}$ and $Δ_{out}$ are the maximum in-degree and out-degree of the graph, resp. We also give a lower bound of $Ω(\min(Δ_{in}/δ,Δ_{out}/δ,\sqrt{m}/δ,m))$ for detecting the $δ$-contributing set of $t$, showing that the simple ApproxContributions algorithm is already optimal. We also investigate the computational complexity of locally estimating a node's PageRank centrality. We improve the best-known upper bound of $\widetilde{O}(n^{2/3}\cdot\min(Δ_{out}^{1/3},m^{1/6}))$ given by Bressan, Peserico, and Pretto (SICOMP '23) to $O(n^{1/2}\cdot\min(Δ_{in}^{1/2},Δ_{out}^{1/2},m^{1/4}))$ by simply combining ApproxContributions with the Monte Carlo simulation method. We also improve their lower bound of $Ω(\min(n^{1/2}Δ_{out}^{1/2},n^{1/3}m^{1/3}))$ to $Ω(n^{1/2}\cdot\min(Δ_{in}^{1/2},Δ_{out}^{1/2},m^{1/4}))$ if $\min(Δ_{in},Δ_{out})=Ω(n^{1/3})$, and to $Ω(n^{1/2-γ}(\min(Δ_{in},Δ_{out}))^{1/2+γ})$ if $\min(Δ_{in},Δ_{out})=o(n^{1/3})$, where $γ>0$ is an arbitrarily small constant. Our matching upper and lower bounds resolve the open problem of whether one can tighten the bounds given by Bressan, Peserico, and Pretto (FOCS '18, SICOMP '23). Remarkably, the techniques and analyses for proving all our results are surprisingly simple.

Figures (3)

• Figure 1: The graph $H$ as a hard instance for detecting the $\delta$-contributing set.
• Figure 2: The multi-level structure.
• Figure 3: The graph $H_i$ for $0\le i\le p$ as hard instances for estimating single-node PageRank. Node $u_{*}$ has $i/p\cdot|\mathcal{Y}|$ parents in $\mathcal{Y}$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Informal
• Theorem 1.4
• Theorem 1.5: Informal
• Theorem 1.6: Informal
• Lemma 2.1: Invariant andersen2008local
• Theorem 2.2: andersen2008local
• Theorem 2.3: lofgren2013personalizedwang2020personalized
• proof : Proof of Theorem \ref{['thm:cost_bp']}