Periods of Connected Networks and Powers of Nonnegative Matrices

Let P be an irreducible N × N matrix having nonnegative entries, and consider the directed network containing arc i, j if and only if Pij is positive. Theorem 1 expresses the period of this network in terms of any of its arborescences. Theorem 2 shows that if the network's period is one and if node i is contained in a cycle of length n, then the ith row and column of Pt are positive for t ≥ N-1n. Theorem 3 shows how to compute the network's period with work proportional to the number of its arcs.