Max-Balancing Weighted Directed Graphs and Matrix Scaling

A weighted directed graph G IS a triple (V, A . g) where (V. A) IS a directed graph and g is a n arbitrary real-valued function defined on the arc set A. Let G be a strongly-connected, simple weighted directed graph. We say tha t G is max-balanced if fo r every nontrivial ~ ubset of the vertices W, the maxImum weight over arcs leaving W equals the maximum weIght over arcs entering W. We show that there exists a (up to an addItIve con~tant) unique potential p, for ( E V such that (V, A, g") IS max-balanced where g/: = P" + go PI for a = (U , I ) EA. We describe an O(1V1 IAI) algorithm for computlJ1g P using an a lgorithm for computing the tnaxmwm cycle-mean of C. Fmally. we apply our principal result to the similarity scaling of nonnegatIve matrices.