Multi-Scale Matrix Sampling and Sublinear-Time PageRank Computation

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 $Δ$, and no node of PageRank smaller than $Δ/c$. We call this problem {\sc SignificantPageRanks}. We develop a nearly optimal, local algorithm for the problem with runtime complexity $\tilde{O}(n/Δ)$ on networks with $n$ nodes. We show that any algorithm for solving this problem must have runtime of $Ω(n/Δ)$, 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 $ε$. 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 any column whose sum is less than $Δ/c$. Our multi-scale sampling scheme solves this problem with cost $\tilde{O}(n/Δ)$, 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 \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.

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