This paper examines a multiple time-scale system where each layer of time-scale dynamics corresponds to a network. Coupling between layers induces a network-of-networks dynamic system. Assuming a hierarchical interaction structure between network time-scale layers with sufficient time-scale differences, layers can be studied under a separation principle. We describe stability of the network-of-networks through a composite Lyapunov function and provide a bilinear matrix inequality condition to guarantee asymptotic stability. Techniques are proposed to adapt this inequality into a linear matrix inequality condition making it computationally efficient to solve as a convex optimization problem. We examine estimates of the dynamics’ domain of attraction and provide conservative bounds on the time-scale separation required to guarantee stability. A class of network-of-networks dynamics with modified consensus dynamics and a state-dependent network structure is then explored. Graph-based guarantees are provided that certify the stability of the system with the worst case rate of convergence dictating conservative bounds on the time-scale separation. An illustrative network-of-networks example is presented.