I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but we have two different islands in bipartite graph and connections occur between nodes from different groups. 



Stationary distribution for bipartite graph
There is a distribution such that, if you start a random walk in that distribution, then it will remain there for all time: the usual formula, $\pi(v) = d(v)/2e(G)$, still works. What is no longer true is that the random walk will converge to the stationary distribution as time tends to infinity independent of the start vertex, as there is an unsurmountable parity problem: if you start in one class, then you will always be in that class after an even number of steps, so the probability of being at a particular vertex is zero at every other time step. In the language of Markov chains, the random walk on a bipartite graph is *periodic*.


