One of the major challenges in systems and synthetic biology is the lack of modular composition. Modules change their behavior once connected, due to retroactivity. In this paper, we build upon our earlier results and provide a theorem establishing how the dynamics of a master module change once slave modules are present. We quantify the change in the dynamics of the master module due to interconnection as a function of measurable biochemical parameters. Based on this, we provide a bound on the difference between the trajectories of the connected system and those of the isolated system by employing contraction theory. Therefore, we obtain a measure of robustness, which helps evaluating the degree of modularity in a system, while providing guidelines for robust module design. We illustrate the results by considering a recurring motif in gene transcription networks: An autorepressed gene regulating the expression of several downstream targets.