Quantifying the Controllability of Coarsely Characterized Networked Dynamical Systems

We study the controllability of large-scale networked dynamical systems when complete knowledge of network structure is unavailable. In particular, we establish the power of learning community-based representations to understand the ability of a group of control nodes to steer the network to a target state. We are motivated by abundant real-world examples, ranging from power and water systems to brain networks, in which practitioners do not have access to fine-scale knowledge of the network. Rather, knowledge is limited to coarse summaries of network structure. Existing work on "model order reduction" starts with full knowledge of fine-scale structure and derives a coarse-scale (lower-dimensional) model that well-approximates the fine-scale system. In contrast, in this paper the controllability aspects of the coarse system are derived from coarse summaries {\em without} knowledge of the fine-scale structure. We study under what conditions measures of controllability for the (unobserved) fine-scale system can be well approximated by measures of controllability derived from the (observed) coarse-scale system. To accomplish this, we require knowledge of some inherent parametric structure of the fine-scale system that makes this type of inverse problem feasible. To this end, we assume that the underlying fine-scale network is generated by the stochastic block model (SBM) often studied in community detection. We quantify controllability using the ``average controllability'' metric and bound the difference between the controllability of the fine-scale system and that of the coarse-scale system. Our analysis indicates the necessity of underlying structure to make possible the learning of community-based representations, and to be able to quantify accurately the controllability of coarsely characterized networked dynamical systems.