Transcription of 5 Random Walks and Markov Chains
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5 Random Walks and Markov Chains
5 RandomWalksand Markov ChainsA randomwalk on a directedgraphconsistsof a sequenceof verticesgeneratedfroma startvertexby selectingan edge,traversingthe edgeto a newvertex,andrepeatingthe will see thatif the graphis stronglyconnected,thenthe fractionof timethe walk spendsat the variousverticesof the graphconvergesto a graphis directed,theremight be verticeswithno out edgesandhencenowherefor the walk to go. Verticesin a stronglyconnectedcomponent withno in edgesfromthe remainderof the graphcan never be reached unlessthe component walk leaves a stronglyconnectedcomponent it can never our discussionof randomwalkswill involve Random walk at a vertexx0and thinkof the startingprobability distributionas putting a massof one onx0andzero on every , onecouldstart withany probability distributionp, wherepis a row vectorwithnonnegativecomponents summingto one,withpxbeingthe probability of startingat vertexx.
undirected graph time reversible Table 5.1: Correspondence between terminology of random walks and Markov chains analogy between random walks and electrical networks. Aspects of the theory of random walks was developed in computer science with an important application in defining the pagerank of pages on the World Wide Web by their
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