On limits of powers of certain absolutely row-stochastic matrices

We consider limits of powers of matrices that are absolutely row-stochastic matrices A such that vertical bar A vertical bar is row-stochastic. We give graph theoretic criteria when such limits exist, and if so, determine their value. In particular, we show that for aperiodic connected matrices that satisfy '+-opposition bipartiteness' (+OBIPness) a 'Perron-Frobenius property' holds. Here, we call A +OBIP (also called structural balance in the literature) when nodes in the graph corresponding to A can be partitioned into two groups such that within-group links are non-negative and across-group links are non-positive. (C) 2016 Elsevier Inc. All rights reserved.