Causal discovery from observational data remains a challenge in the study of the physics of complex systems, where causal relationships must be inferred without controlled interventions or prior knowledge of the underlying mechanisms. A major obstacle is the presence of hidden common causes, which can induce statistical dependencies between variables and thereby confound causal claims.
Here, we discuss a causal discovery framework rooted in control theory and network science that offers a promising approach to addressing this issue in stochastic dynamical systems on complex networks.
The framework involves estimating network size from the dynamics of a few nodes, a problem that has recently attracted attention in the physics community and is also addressed in this work.
We present the theoretical formulation of the framework and provide an operational estimation procedure to demonstrate its applicability on real data.

We first validate the control theoretic approach on synthetic dynamical systems on random networks.
We then exemplify the framework on a network of crypto-assets, a complex system driven by hidden exogenous forces, where the reconstructed graph identifies Bitcoin as a root node driving most other assets.
Finally, we address a long-standing debate in opinion dynamics on what shapes public sentiment toward the police where we find that both media coverage of police misconduct and local crime
drive public sentiment toward the police.