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.

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 develop a new bidirectional algorithm for estimating Markov chain multi-step transition probabilities: given a Markov chain, we want to estimate the probability of hitting a given target state in $\ell$ steps after starting from a given source distribution. Given the target state $t$, we use a (reverse) local power iteration to construct an `expanded target distribution', which has the same mean as the quantity we want to estimate, but a smaller variance -- this can then be sampled efficiently by a Monte Carlo algorithm. Our method extends to any Markov chain on a discrete (finite or countable) state-space, and can be extended to compute functions of multi-step transition probabilities such as PageRank, graph diffusions, hitting/return times, etc. Our main result is that in `sparse' Markov Chains -- wherein the number of transitions between states is comparable to the number of states -- the running time of our algorithm for a uniform-random target node is order-wise smaller than Monte Carlo and power iteration based algorithms; in particular, our method can estimate a probability $p$ using only $O(1/\sqrt{p})$ running time.

Latency is a critical factor when using a crowdsourcing platform to solve a problem like entity resolution or sorting. In practice, most frameworks attempt to reduce latency by heuristically splitting a budget of questions into rounds, so that after each round the answers are analyzed and new questions are selected. We focus on one of the most extensively studied crowdsourcing operations, the MAX operation (finding the best element in a collection under human criteria), and we study the problem of budget allocation into rounds for this operation. We provide a polynomial-time dynamic-programming budget allocation algorithm that minimizes the latency when questions form tournaments in each round. Furthermore, we study the general case where questions can be asked in any arbitrary way in each round. Our theoretical results for the general case indicate that our approach is also optimal under certain worst and average-case scenarios. We compare our approach to alternatives on Amazon Mechanical Turk, where many of our theory assumptions do not necessarily hold. We find that our approach is also optimal in practice and achieves a notable improvement over alternatives in most cases.

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, it computes the Personalized PageRank π_s(t) from s to t, guaranteeing that the relative error is small 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 running-time 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 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 difficult 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 efficient 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.