  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.
