Employing the framework of integral input-to-state stability (iISS), this paper studies the problem of verifying stability of dynamical networks. The iISS we impose on the subsystems encompasses a wider variety of nonlinearities than input-to-state stability (ISS) which has been studied extensively in the literature. To go beyond the ISS results, this paper investigates a simultaneous small-gain criterion, a topological separation criterion and a spectral radius formula in view of necessity and suffiency for the stability of the iISS network, and they are related to each other. This work aims at unifying conceivable stability criteria into a single methodology in the iISS formulation which includes ISS as a special case. For illustration, this paper examines the dynamics of network computing for resource utilization. Based on a fluid flow model, the proposed iISS methodology provides us with qualitative and quantitative information on the computing speed, the communication overhead, the number of nodes and the interconnection structure for achieving successful workload distribution across multiple computers.