Multi-Scale Matrix Sampling and Sublinear-Time PageRank Computation

• Published in: arXiv.org
• Main Authors: Borgs, Christian, Brautbar, Michael, Chayes, Jennifer, Shang-Hua Teng
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 28.05.2013
• Subjects: Algorithms; Approximation; Mathematical analysis; Network analysis; Nodes; Optimization; Run time (computers); Sampling; Search engines; Social networks
• ISSN: 2331-8422

A fundamental problem arising in many applications in Web science and social network analysis is, given an arbitrary approximation factor \(c>1\), to output a set \(S\) of nodes that with high probability contains all nodes of PageRank at least \(\Delta\), and no node of PageRank smaller than \(\Delta/c\). We call this problem {\sc SignificantPageRanks}. We develop a nearly optimal, local algorithm for the problem with runtime complexity \(\tilde{O}(n/\Delta)\) on networks with \(n\) nodes. We show that any algorithm for solving this problem must have runtime of \({\Omega}(n/\Delta)\), rendering our algorithm optimal up to logarithmic factors. Our algorithm comes with two main technical contributions. The first is a multi-scale sampling scheme for a basic matrix problem that could be of interest on its own. In the abstract matrix problem it is assumed that one can access an unknown {\em right-stochastic matrix} by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter \(\epsilon\). At a cost propositional to \(1/\epsilon\), the query will return a list of \(O(1/\epsilon)\) entries and their indices that provide an \(\epsilon\)-precision approximation of the row. Our task is to find a set that contains all columns whose sum is at least \(\Delta\), and omits any column whose sum is less than \(\Delta/c\). Our multi-scale sampling scheme solves this problem with cost \(\tilde{O}(n/\Delta)\), while traditional sampling algorithms would take time \(\Theta((n/\Delta)^2)\). Our second main technical contribution is a new local algorithm for approximating personalized PageRank, which is more robust than the earlier ones developed in \cite{JehW03,AndersenCL06} and is highly efficient particularly for networks with large in-degrees or out-degrees. Together with our multiscale sampling scheme we are able to optimally solve the {\sc SignificantPageRanks} problem.