Controllability of Coarsely Measured Networked Linear Dynamical Systems (Extended Version)

TL;DR

This paper investigates how to approximate the controllability of large-scale linear networked systems using coarse summaries, especially under stochastic block model assumptions, and provides algorithms and simulations to validate the approach.

Contribution

It introduces a method to estimate fine-scale controllability from coarse network summaries under SBM assumptions, with theoretical analysis and empirical validation.

Findings

Approximate controllability can be accurately estimated from coarse summaries.

Community structure in networks is crucial for controllability approximation.

The method outperforms reduced-order approaches in certain regimes.

Abstract

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.