We present new algorithms for Personalized PageRank estimation and Personalized PageRank search. First, for the problem of estimating Personalized PageRank (PPR) from a source distribution to a target node, we present a new bidirectional estimator with simple yet strong guarantees on correctness and performance, and 3x to 8x speedup over existing estimators in experiments on a diverse set of networks. Moreover, it has a clean algebraic structure which enables it to be used as a primitive for the Personalized PageRank Search problem: Given a network like Facebook, a query like "people named John," and a searching user, return the top nodes in the network ranked by PPR from the perspective of the searching user. Previous solutions either score all nodes or score candidate nodes one at a time, which is prohibitively slow for large candidate sets. We develop a new algorithm based on our bidirectional PPR estimator which identifies the most relevant results by sampling candidates based on their PPR; this is the first solution to PPR search that can find the best results without iterating through the set of all candidate results. Finally, by combining PPR sampling with sequential PPR estimation and Monte Carlo, we develop practical algorithms for PPR search, and we show via experiments that our algorithms are efficient on networks with billions of edges.

We study the problem of enhancing Entity Resolution (ER) with the help of crowdsourcing. ER is the problem of clustering records that refer to the same real-world entity and can be an extremely di cult process for computer algorithms alone. For example, figuring out which images refer to the same person can be a hard task for computers, but an easy one for humans. We study the problem of resolving records with crowdsourcing where we ask questions to humans in order to guide ER into producing accurate results. Since human work is costly, our goal is to ask as few questions as possible. We propose a probabilistic framework for ER that can be used to estimate how much ER accuracy we obtain by asking each question and select the best question with the highest expected accuracy. Computing the expected accuracy is #P-hard, so we propose approximation techniques for e cient computation. We evaluate our best question algorithms on real and synthetic datasets and demonstrate how we can obtain high ER accuracy while significantly reducing the number of questions asked to humans.

We propose and analyze two algorithms for maintaining approximate Personalized PageRank (PPR) vectors on a dynamic graph, where edges are added or deleted. Our algorithms are natural dynamic versions of two known local variations of power iteration. One, Forward Push, propagates probability mass forwards along edges from a source node, while the other, Reverse Push, propagates local changes backwards along edges from a target. In both variations, we maintain an invariant between two vectors, and when an edge is updated, our algorithm first modifies the vectors to restore the invariant, then performs any needed local push operations to restore accuracy.For Reverse Push, we prove that for an arbitrary directed graph in a random edge model, or for an arbitrary undirected graph, given a uniformly random target node t, the cost to maintain a PPR vector to t of additive error ε as k edges are updated is O(k + d/ε), where d is the average degree of the graph. This is O(1) work per update, plus the cost of computing a reverse vector once on a static graph. For Forward Push, we show that on an arbitrary undirected graph, given a uniformly random start node s, the cost to maintain a PPR vector from s of degree-normalized error ε as k edges are updated is O(k + 1/ε), which is again O(1) per update plus the cost of computing a PPR vector once on a static graph.

Abstract. 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 worstcase 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.

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We propose a new algorithm, FAST-PPR, for computing personalized PageRank: given start node s and target node t in a directed graph, and given a threshold δ, FAST-PPR computes the Personalized PageRank πs(t) from s to t, guaranteeing a small relative error as long πs(t) > δ. Existing algorithms for this problem have a running-time of Ω(1/δ); in comparison, FAST-PPR has a provable average runningtime guarantee of O( d/δ) (where d is the average in-degree of the graph). This is a significant improvement, since δ is often O(1/n) (where n is the number of nodes) for applications. We also complement the algorithm with an Ω(1/ √ δ) lower bound for PageRank estimation, showing that the dependence on δ cannot be improved.We perform a detailed empirical study on numerous massive graphs, showing that FAST-PPR dramatically outperforms existing algorithms. For example, on the 2010 Twitter graph with 1.5 billion edges, for target nodes sampled by popularity, FAST-PPR has a 20 factor speedup over the state of the art. Furthermore, an enhanced version of FAST-PPR has a 160 factor speedup on the Twitter graph, and is at least 20 times faster on all our candidate graphs.