The paper presents a system theoretic analysis framework for a network-of-networks, formed from smaller factor networks via graph Cartesian products. We provide a compositional framework for extending the controllability and observability of the factor networks to that of the composite network-of-networks. We then delve into the effectiveness of designing control and estimation algorithms for the composite network via its symmetry and gramian structure. An example demonstrating the usefulness of our results in the context of social networks with a Cartesian product structure is then presented.