Revisiting Local Computation of PageRank: Simple and Optimal

Hanzhi Wang, Zhewei Wei, Ji-Rong Wen, Mingji Yang

Abstract: We revisit ApproxContributions, the classic local graph exploration algorithm proposed by Andersen, Borgs, Chayes, Hopcroft, Mirrokni, and Teng (WAW ’07, Internet Math. ’08) for computing an є-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 it as O(nπ(t)/є·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 t’s δ-contributing set, showing that the simple ApproxContributions algorithm is already optimal.As ApproxContributions has become a cornerstone for computing random-walk probabilities, our results and techniques can be applied to derive better bounds for various relevant problems. In particular, we 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 combining ApproxContributions with Monte Carlo sampling. 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) otherwise, 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.