A broad set of empirical phenomenon in the study of social, economic and machine behaviour can be modelled as complex systems with averaging dynamics. However many of these models naturally result in consensus or consensus-like outcomes. In reality, empirical phenomenon rarely converge to these and instead are characterized by rich, persistent variation in the agent states. Such heterogeneous outcomes are a natural consequence of a number of models that incorporate external perturbation to the otherwise convex dynamics of the agents. The purpose of this paper is to formalize the notion of heterogeneity and demonstrate which classes of models are able to achieve it as an outcome, and therefore are better suited to modelling important empirical questions. We do so by determining how the topology of (time-varying) interaction networks restrict the space of possible steady-state outcomes for agents, and how this is related to the study of random walks on graphs. We consider a number of intentionally diverse examples to demonstrate how the results can be applied.