Controllability of Coarsely Measured Networked Linear Dynamical Systems (Extended Version)
This work addresses the challenge of analyzing controllability in complex networks with limited data, which is incremental as it builds on existing methods like reduced-order models and community detection techniques.
The paper tackles the problem of approximating controllability in large-scale networked linear dynamical systems when only coarse network summaries are available, by showing that under certain structural assumptions like the stochastic block model, average controllability can be accurately estimated from coarse data, with simulations confirming theoretical results for various network sizes and densities.
We consider the controllability of large-scale linear networked dynamical systems when complete knowledge of network structure is unavailable and knowledge is limited to coarse summaries. We provide conditions under which average controllability of the fine-scale system can be well approximated by average controllability of the (synthesized, reduced-order) coarse-scale system. To this end, we require knowledge of some inherent parametric structure of the fine-scale network that makes this type of approximation possible. Therefore, we assume that the underlying fine-scale network is generated by the stochastic block model (SBM) -- often studied in community detection. We then provide an algorithm that directly estimates the average controllability of the fine-scale system using a coarse summary of SBM. Our analysis indicates the necessity of underlying structure (e.g., in-built communities) to be able to quantify accurately the controllability from coarsely characterized networked dynamics. We also compare our method to that of the reduced-order method and highlight the regimes where both can outperform each other. Finally, we provide simulations to confirm our theoretical results for different scalings of network size and density, and the parameter that captures how much community-structure is retained in the coarse summary.