Network identification with latent nodes via auto-regressive models

We consider linear time-invariant networks with unknown interaction topology where only a subset of the nodes, termed manifest, can be directly controlled and observed. The remaining nodes are termed latent and their number is also unknown. Our goal is to identify the transfer function of the manifest subnetwork and determine whether interactions between manifest nodes are direct or mediated by latent nodes. We show that, if there are no inputs to the latent nodes, then the manifest transfer function can be approximated arbitrarily well in the $H_infty}$-norm sense by the transfer function of an auto-regressive model. Motivated by this result, we present a least-squares estimation method to construct the auto-regressive model from measured data. We establish that the least-squares matrix estimate converges in probability to the matrix sequence defining the desired auto-regressive model as the length of data and the model order grow. We also show that the least-squares auto-regressive method guarantees an arbitrarily small $H_infty$-norm error in the approximation of the manifest transfer function, exponentially decaying once the model order exceeds a certain threshold. Finally, we show that when the latent subnetwork is acyclic, the proposed method achieves perfect identification of the manifest transfer function above a specific model order as the length of the data increases. Various examples illustrate our results.