. Markov Chains and Random Walks on Graphs

∑ j∈S π j p ji = ∑ j∈S π i p i j = π i ∑ j∈S p ji = π i. 2 Observe the intuition underlying the detailed balance condition: At stationarity, an equal amount of probability mass flows in each step from i to j as from j to i.(The " ergodic flows " ' between states are in pairwise balance; cf. Figure 8.) Example 1.6 Random walks on graphs. Define a Markov chain on the nodes of G so that at each step, one of the current node's neigbours is selected as the next state, uniformly at random. Let us check that this chain is reversible, with stationary distribution π = d 1 d d 2 d · · · d n d , where d = ∑ n i=1 d i = 2|E|. The detailed balance condition is easy to verify: Consider the three-state Markov chain shown in Figure 9. It is easy to verify that this chain has the unique stationary distribution π = 1 3 1 3 1 3. However, for any i = 1, 2, 3: π i p i,(i+1) = 1 3 · 2 3 = 2 9 > π i+1 p (i+1),i = 1 3 · 1 3 = 1 9. Thus, even in a stationary situation, the chain has a " preference " of moving in the counterclockwise direction, i.e. it is not time-symmetric.