Some extremal Markov chains

Using a Markov chain model for the motion of a particle through a V-node network, we consider the quantities nij, which are the average number of steps taken by the particle in traveling from an originating node, i, to a destination node, j. A figure-of-merit, N, for the entire network is introduced by averaging nij over i and j. We investigate which networks minimize or maximize N, either when no restriction is placed on the Markov chain, or when we restrict it so that it corresponds to random routing. By the latter we mean that at each node the particle “selects at random” lines from an undirected network graph. We show that for random routing, the complete graph has N = (V − 1) and is the minimizing graph. The maximizing graph is unknown, but we establish that the worst behavior of N increases at least with the cube of the number of vertices, but no worse than the 3.5 power. Properties of the class of graphs known as barbells are useful here. The minimizing unrestricted chain corresponds to placing the nodes on a circle and proceeding unidirectionally from one node to the next. Here, N = V/2.