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which becomes the uniforn distribution on &2 when p- 112 Exercise 2 (Random walk on disconnected graphs). Let G - (V,E) be a graph with two disjoint components Gı and G2. Let (Xt)tz0 be a random walk on G and denote by P its transition matrix. (i) Let Pi and P2 be the transition matrices for random walks on Gı and G2, respectively. Let Vi = {1, 2, , n} and ½ = {n + 1, n + 2, , n + m} be the set of nodes in Gl and G2. Then show that P is of the following block diagonal form Pi O O P2 (i) Let π1= [π11), . . . π(n)] and π2= [π(1), ,π j be any stationary distributions forP, and P2. Let be distributions on V-V1 U -(1, 2, n, n + 1, , n + m. Show that for any λ ε [0, 1], π :Ξ λ 1 (1-1)Tạis a stationary distribution for P. (iii) Recall that random walk on arbitrary graph has a stationary distribution, where probability of each node is proportional to its degree. Conclude that the random walk (Xt)tz20 on G has infinitelv manv stationarv distribution.

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