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Unformatted text preview: April 19, 2010 Wensong(Tony) Xu Eigenvalue, Diameter, and Polynomial Convergence 1 Introduction In this lecture, we will show the polynomial convergence of a random walk on a non-bipartite undirected graph, and its connection with the second largest eigenvalue and the diameter of the graph. 2 Polynomial Convergence Let G be a non-bipartite undirected graph, and P be the transition matrix of a random walk on G . Define Δ( s ) = max x,y | P s ( y,x )- π ( x ) | π ( x ) Then we have Δ( s ) ≤ e- c if s ≥ 1 λ (log Vol G min d x + c ) where λ = λ 1 if λ 1 ≤ 2- λ n- 1 , 2- λ n- 1 otherwise. Note that only λ 1 matters here, since we can always transform the random walk to a lazy random walk by letting P = 1 2 ( I + P ) , then λ 1 = 1 2 λ 1 and λ n- 1 becomes less than 1 . Hence if 1 λ 1 is polynomial, then the convergence is polynomial. Exercise 1. What G with minimum value of λ 1 , among all connected graph with n vertices? (Hint: two mass subgraph with 1 edge in between.) Note 1. For a k regular expander graph, 1- λ 1 ∼ 2 √ k- 1 k Note 2. There is no polynomial bound on convergence of random walk for directed graph, even the graph...
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This note was uploaded on 09/30/2011 for the course MATH 262 taught by Professor Aterras during the Fall '08 term at UCSD.
- Fall '08