The paper presents an analysis framework for a class of dynamic composite networks. These networks are formed from smaller factor networks via graph Cartesian products. We provide a composition method for extending the controllability and observability of the factor networks to that of the composite network. We then delve into the effectiveness of designing control and estimation algorithms for the composite network via symmetry in the network. Examples and applications are provided throughout the paper to demonstrate the results including a distributed output feedback stabilizer and a social network application.