Transcription of 5 Random Walks and Markov Chains
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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.
of random walks and Markov chains is given in Table 5.1. A state of a Markov chain is persistent if it has the property that should the state ever be reached, the random process will return to …
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