Revisiting Local Computation of PageRank: Simple and Optimal

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·min(Δᵢₙ,Δₒᵤₜ,√m)), where π(t) is the PageRank score of t, and Δᵢₙ and Δₒᵤₜ are the maximum in-degree and out-degree of the graph, resp. We also give a lower bound of Ω(min(Δᵢₙ/δ,Δₒᵤₜ/δ,√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 O(n2/3·min(Δₒᵤₜ1/3,m1/6)) given by Bressan, Peserico, and Pretto (SICOMP '23) to O(n1/2·min(Δᵢₙ1/2,Δₒᵤₜ1/2,m1/4)) by simply combining ApproxContributions with the Monte Carlo simulation method. We also improve their lower bound of Ω(min(n1/2Δₒᵤₜ1/2,n1/3m1/3)) to Ω(n1/2·min(Δᵢₙ1/2,Δₒᵤₜ1/2,m1/4)) if min(Δᵢₙ,Δₒᵤₜ)=Ω(n1/3), and to Ω(n1/2-γ(min(Δᵢₙ,Δₒᵤₜ))1/2+γ) if min(Δᵢₙ,Δₒᵤₜ)=o(n1/3), 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.