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 |A||A| 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.