Lecture5

# Lecture5 - We are interested in studying random walks in...

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We are interested in studying random walks in graphs by using the matrix P . Examples of real life phenomena that can be modeled by random walks include the orderings of a deck of cards or state graphs. A graph is ergodic if there exists a unique stationary distribution, Π, such that lim s →∞ fP s ( v ) = Π( v ) where f is the initial probability distribution among the vertices. There are two necessary conditions for the random walks to be Markov chains. These conditions also end up being suﬃcient. The graph must be irreducible, meaning that for all pairs of vertices u and v , there exists an in- teger s such that P s ( u,v ) > 0. This means that there exists a walk between any two vertices. The graph must also be aperiodic. This means that for all pairs of vertices u and v , gcd { s : P s ( u,v ) > 0 } = 1. This means that the lengths of all the walks from u to v have relatively prime length. For undi- rected graphs, irreducibility is equivalent to connectivity, and aperiodicity is

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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.

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Lecture5 - We are interested in studying random walks in...

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