P(x,y) = desca) A(x, y), T = = Random walk on a simple undirected graph G (V, E) is a reversible Markov chain. To see this, recall that in this case the transition matrix P satisfies 1 (x where A is the adjacency matrix. Define a distribution a that is proportional to the degree distribution, as follows: deg(a) (x): 2 EL We can then check that this a satisfies the balance equation, because 1 7 (2)P(x, y) = (y)P(y,x) A(x,y). 2 E| (b) Consider the cycle graph Cn on N nodes, labeled consecutively as X0, X1, ..., XN, where xn = xo. Let {Xn}"-o be a biased random walk on Cn with transition ma- trix given by P(Xk, Xk+1) Vk = 0,...,N 1, 4' Let A be the event that Xn starts at wo and then goes around Cn once in one direction. Namely, assume A = {Xo = x0, X1 = x1, ..., Xn-1 = {N-1, Xn = xo}. Compute the probability of this event, then compare it to the event that the walk goes around Cn in the reverse order. What do you notice? Is this chain reversible? 2 a 1 = = = = = P(x,y) = desca) A(x, y), T = = Random walk on a simple undirected graph G (V, E) is a reversible Markov chain. To see this, recall that in this case the transition matrix P satisfies 1 (x where A is the adjacency matrix. Define a distribution a that is proportional to the degree distribution, as follows: deg(a) (x): 2 EL We can then check that this a satisfies the balance equation, because 1 7 (2)P(x, y) = (y)P(y,x) A(x,y). 2 E| (b) Consider the cycle graph Cn on N nodes, labeled consecutively as X0, X1, ..., XN, where xn = xo. Let {Xn}"-o be a biased random walk on Cn with transition ma- trix given by P(Xk, Xk+1) Vk = 0,...,N 1, 4' Let A be the event that Xn starts at wo and then goes around Cn once in one direction. Namely, assume A = {Xo = x0, X1 = x1, ..., Xn-1 = {N-1, Xn = xo}. Compute the probability of this event, then compare it to the event that the walk goes around Cn in the reverse order. What do you notice? Is this chain reversible? 2 a 1 = = = = =