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Exploring Mixing Time in Lazy Chains and Random Walks on Graphs

This seminar presentation covers the analysis of mixing time for lazy Markov chains within the framework of random walks on graphs. It outlines crucial proofs and methodologies, including bounding distances to stationary distributions and inductive arguments to derive mixing time bounds. The discussion includes the properties of piece-wise linear functions, breaks, and the roles of various graph vertices. Through structured proofs and intuition, the presentation elucidates the relationship between transition probabilities and convergence to equilibrium.

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Exploring Mixing Time in Lazy Chains and Random Walks on Graphs

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  1. Mixing Time – General Chains Seminar on Random Walks on Graphs 2009/2010 Ilan Ben Bassat Omri Weinstein

  2. Mixing Time - Lazy Chains Given that: Then:

  3. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  4. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  5. Bounding Distributions Distance For every

  6. So, choosing for every S yields: And for every vertex w we can get:

  7. H(x) Values Order the graph vertices: And define . Find k such that: Then the value of ht(x) is obtained by: And we get:

  8. Function H - Analysis • Piece-wise linear function. • Concave • Breakpoints at • On the interval [0,1]: and

  9. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  10. Proof Sketch and Intuition We will prove for every x and t: Intuition: Bound the value h(x) in time t by going one step backwards, and a bit towards the endpoints. Prove will be in three parts: • For a finite group of discrete x values . • For all x values satisfying (and symmetric case). • For all x values and

  11. Proof – Breakpoints Fix k and let

  12. Proof – cont’d

  13. Proof – second case

  14. Proof – Third Part So, for every x we get:

  15. Base Case

  16. Induction For t=0 trivial. For and

  17. Induction

  18. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  19. Mixing Time Proof Given

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