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An Overview of Markov Chains: Theory and Applications in Network and Matrix Computations

This presentation by David Gleich from Purdue University delves into the fundamentals of Markov chains, covering topics such as transient and recurrent states, ergodicity, the Perron-Frobenius theorem, and stationary distributions. Key applications like PageRank and centrality measures in spectral graph theory are discussed, alongside the importance of stochastic processes and random walks. The outline includes discussions on hitting and commute times, transition-related problems, and significant theoretical insights that underpin the analysis of Markov processes.

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An Overview of Markov Chains: Theory and Applications in Network and Matrix Computations

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Presentation Transcript

  1. Overview of Markov chains David Gleich Purdue University Network & Matrix Computations Computer Science 15 Sept 2011

  2. Lots of words transient recurrent primitive regular ergodic periodic irreducible Perron root Perron vector stationary distribution first transition analysis Cesáro limit reversible Markov chain simple stationary distribution Perron-Frobenius theorem

  3. Some theory • Theorem Let A ≥ 0 be irreducible then A has a unique positive eigenvector with eigenvalue equal to the spectral radius of A.

  4. In German Oskar Perron, MathematischeAnnalen, 64:248-263, 1907 doi: 10.1007/BF01449896

  5. Just kidding

  6. Important application • PageRank • centrality measures • spectral graph theory • random sampling

  7. Probable outline • Stochastic processes, Markov chains, and random walks, and stochastic matrices • The Perron-Frobenius theorem and stationary distribution • Hitting times, commute times, and other transition related problems

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