Multiscale Matrix Sampling and Sublinear-Time PageRank Computation

Abstract

A fundamental problem arising in many applications in Web science and social network analysis is the problem of identifying all nodes in a network whose PageRank exceeds a given threshold Δ. In this paper, we study the probabilistic version of the problem whereby given an arbitrary approximation factor _c_ > 1, we are asked to output a set _S_ of nodes such that with high probability, _S_ contains all nodes of PageRank at least Δ, and no node of PageRank smaller than Δ/_c_. We call this problem SignificantPageRanks. We develop a nearly optimal local algorithm for the problem with time complexity <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_802752_ilm0001.gif"></inline-graphic></inline-formula> on networks with _n_ nodes, where the tilde hides a polylogarithmic factor. We show that every algorithm for solving this problem must have running time of Ω(_n_/Δ), rendering our algorithm optimal up to logarithmic factors. Our algorithm has sublinear time complexity for applications including Web crawling and Web search that require efficient identification of nodes whose PageRanks are above a threshold Δ = _n_^δ, for some constant 0 < δ < 1. Our algorithm comes with two main technical contributions. The first is a multiscale sampling scheme for a basic matrix problem that could be of interest on its own. For us, it appears as an abstraction of a subproblem we need to tackle in order to solve the SignificantPageRanks problem, but we hope that this abstraction will be useful in designing fast algorithms for identifying nodes that are significant beyond PageRank measurements. In the abstract matrix problem, it is assumed that one can access an unknown _right-stochastic matrix_ by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter ε. At a cost propositional to 1/ε, the query will return a list of _O_(1/ε) entries and their indices that provide an ε-precision approximation of the row. Our task is to find a set that contains all columns whose sum is at least Δ and omits every column whose sum is less than Δ/_c_. Our multiscale sampling scheme solves this problem with cost <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_802752_ilm0002.gif"></inline-graphic></inline-formula>, while traditional sampling algorithms would take time Θ((_n_/Δ)²). Our second main technical contribution is a new local algorithm for approximating personalized PageRank, which is more robust than the earlier ones developed in [Jeh and Widom 03, Andersen et&nbsp;al. 06] and is highly efficient, particularly for networks with large in-degrees or out-degrees. Together with our multiscale sampling scheme, we are able to solve the SignificantPageRanks problem optimally.