Bidirectional PageRank Estimation: From Average-Case to Worst-Case

We present a new algorithm for estimating the Personalized PageRank (PPR) between a source and target node on undirected graphs, with sublinear running-time guarantees over the worst-case choice of source and target nodes. Our work builds on a recent line of work on bidirectional estimators for PPR, which obtained sublinear running-time guarantees but in an average-case sense, for a uniformly random choice of target node. Crucially, we show how the reversibility of random walks on undirected networks can be exploited to convert average-case to worst-case guarantees. While past bidirectional methods combine forward random walks with reverse local pushes, our algorithm combines forward local pushes with reverse random walks. We also discuss how to modify our methods to estimate random-walk probabilities for any length distribution, thereby obtaining fast algorithms for estimating general graph diffusions, including the heat kernel, on undirected networks.Comment: Workshop on Algorithms and Models for the Web-Graph (WAW) 201

Approximating PageRank locally with sublinear query complexity

The problem of approximating the PageRank score of a node with minimal information about the rest of the graph has attracted considerable attention in the last decade; but its central question, whether it is in general necessary to explore a non-vanishing fraction of the graph, remained open until now (only for specific graphs and/or nodes was a solution known). We present the first algorithm that produces a (1\pm\epsilon)-approximation of the score of any one node in any n-node graph with probability (1-\epsilon) visiting at most O(n^\frac{2}{3}\sqrt[3]{\log(n)})=o(n) nodes. Our result is essentially tight (we prove that visiting \Omega(n^\frac{2}{3}) nodes is in general necessary to solve even an easier "ranking" version of the problem under any "natural" graph exploration model, including all those in the literature) but it can be further improved for some classes of graphs and/or nodes of practical interest - e.g. to O(n^\frac{1}{2} \gamma^\frac{1}{2}) nodes in graphs with maximum outdegree \gamma

Brief announcement: On approximating pageRank locally with sublinear query complexity

Can one compute the pageRank score of a single, arbitrary node in a graph, exploring only a vanishing fraction of the graph? We provide a positive answer to this extensively researched open question. We develop the first algorithm that, for any n-node graph, returns a multiplicative (1±ϵ)-approximation of the score of any given node with probability (1−δ), using at most On2/3ln(n)1/3ln(1/δ)2/3ϵ−2/3= Õ(n2/3) queries which return either a node chosen uniformly at random, or the list of neighbours of a given node. Alternatively, we show that the same guarantees can be attained by fetching at most OE4/5d−3/5ln(n)1/5ln(1/δ)3/5ϵ−6/5= Õ(E4/5) arcs, where E is the total number of arcs in the graph and d is its average degree