This project aims to formulate a mathematical theory to shed light on emergent behavior in complex networks of non-linear dynamic systems. Examples of such complex networks include the brain, power grids, social networks, protein networks, and sensors in smart cities. These networks exhibit a mixed global behavior, with significant phenomena occurring on finite time scales. Consequently, conventional tools are inadequate for studying these systems. This proposal seeks to bridge this gap by developing a theory for emergent phenomena in complex networks. This could unlock vast potential, such as the ability to reconstruct system rules based on their behavior. If successful, this research could enable the prediction of critical transitions and the prevention of catastrophes.