This paper proposes a methodology for establishing stability of nonlinear dynamical networks through balancing mechanisms that appear in system models explicitly. In addition to global asymptotic stability of equilibria, the notion of strong integral input-to-state stability (Strong iISS) is employed to address robustness of the networks with respect to disturbances and noises in the presence of saturation nonlinearities forming balancing relations. Conservation is covered as the exact balance. Unlike network small-gain criteria known in the literature of iISS, the developed criteria do not lump the balancing relations. The formulation covers positive networks and networks of iISS systems.