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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 ( s ) = max x,y | P s ( y,x )- ( x ) | ( x ) Then we have ( s ) e- c if s 1 (log Vol G min d x + c ) where = 1 if 1 2- n- 1 , 2- n- 1 otherwise. Note that only 1 matters here, since we can always transform the random walk to a lazy random walk by letting P = 1 2 ( I + P ) , then 1 = 1 2 1 and n- 1 becomes less than 1 . Hence if 1 1 is polynomial, then the convergence is polynomial. Exercise 1. What G with minimum value of 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- 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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- Fall '08