Network Identification With Latent Nodes via Autoregressive Models

Abstract

We consider linear time-invariant networks with unknown topology where only a manifest subset of the nodes can be directly actuated and measured while the state of the remaining latent nodes and their number are 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, the manifest transfer function can be approximated arbitrarily well in the $H_{\infty }$ -norm sense by the transfer function of an autoregressive model and present a least-squares estimation method to construct the autoregressive model from measured data. We show that the least-squares autoregressive 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.