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Center for Nonlinear Studies |
A fundamental problem in Markov chains is of estimating the probability of transitioning from a given starting state to a given terminal state in a fixed number of steps. This has received much attention in recent years as Markov chains form the basis of many network centrality measures, in particular, PageRank and Personalized PageRank (PPR). Standard approaches to this problem use either linear-algebraic iterative techniques (such as the power iteration) or Monte Carlo - both however have a running time which scales linearly in the size of the network. This is too slow for real-time computation on large networks - consequently, PPR, which has long been recognized as an effective measure for ranking search results, is rarely used in practice. I will present a new approach towards designing bidirectional estimators, which combines linear algebraic and random walk techniques. Our approach provides the first algorithm for PageRank estimation which has sublinear running-time guarantees in theory, and which is much faster than existing algorithms in practice. In particular, we show that it returns estimates with additive error in time in undirected networks, and in sparse directed networks. Our approach extends to general Markov chains, and more generally, to estimating a single element of a linear system.
Host: Harsha Nagarajan