Identification of linear networks with latent nodes

Abstract

We study identification of linear networks under the assumption that only a subset of all the nodes in the network can be observed. The observable nodes are called manifest nodes and they form the manifest subnetwork. The unobservable nodes are called latent nodes and the number of latent nodes is unknown. We explore the possibility of identifying the transfer function of the manifest subnetwork and whether an interaction between two manifest nodes is direct or mediated by latent nodes. In particular, we show that if the external inputs are injected into a linear network only through the manifest nodes, then there exists an auto-regressive model whose transfer function is arbitrarily close to the transfer function of the manifest subnetwork in the H∞ norm sense. Moreover, we prove that the least-squares method provides consistent estimate of the auto-regressive model using the measured states of the observed nodes. Finally, we show that if the latent subnetwork is acyclic, then the transfer function of the manifest subnetwork can be perfectly identified using the least-squares auto-regressive method. Various examples illustrate our results.