Network systems have become a ubiquitous modeling tool in many areas of science where nodes in a graph represent distributed processes and edges between nodes represent a form of dynamic coupling. When a network topology is already known (or partially known), two associated goals are, to derive estimators for nodes of the network, which cannot be directly observed or are impractical to measure; and to quantitatively identify the dynamic relations between nodes. We address both problems in the challenging scenario where only some outputs of the network are being measured and the inputs are not accessible. The approach makes use of the notion of $d$ -separation for the graph associated with the network. In the considered class of networks, it is shown that the proposed technique can determine or guide the choice of optimal sparse estimators. This article also derives identification methods that are applicable to cases where loops are present providing a different perspective on the problem of closed-loop identification. The notion of $d$ -separation is a central concept in the area of probabilistic graphical models, thus an additional contribution is to create connections between control theory and machine learning techniques.

This paper considers the problem of learning the unknown structure of a network with the underlying topology given by a polyforest (a collection of directed trees with potentially multiple roots). The main result is an algorithm that consistently learns the network structure using only second-order statistics of the data. The methodology is robust with respect to the presence of unmeasured (latent) nodes: the algorithm detects the exact number and location of the latent nodes, when they satisfy specific degree conditions in the actual network graph. It is shown that the same degree conditions are also necessary for a consistent reconstruction. Thus, the proposed reconstruction algorithm achieves the fundamental limitations in learning the structure of a polyforest network of linear dynamic systems in the presence of latent nodes. This paper overcomes the limitations of previous results that only addressed single-rooted trees, tackling the problem in an efficient way since the computational complexity of the derived algorithm is proven to be polynomial in the number of observed nodes.

Reconstructing the causal structure of a network of dynamic systems from observational data is an important problem in many areas of science. One of the earliest and most prominent approaches to this problem is Granger causality. It has been shown that in a network with linear dynamics and strictly causal transfer functions, Granger causality consistently reconstructs the underlying graph of the network. On the other hand, techniques that allow for the presence of direct feedthroughs usually assume there are no feedback loops in the dynamics of the network. In this article, we develop an extension of Granger causality that provides theoretical guarantees for the reconstruction of a network topology even in the presence of direct feedthroughs and feedback loops. The only required assumption is a relatively mild condition of well-posedness named recursiveness, where at least one strictly causal transfer function needs to be present in every feedback loop. The technique is compared with other state-of-the-art methods on a benchmark example specifically designed to include several dynamic configurations that are challenging to reconstruct.

This article addresses the problem of consistently identifying a single transfer function in a network of dynamic systems using only observational data. It is assumed that the topology is partially known, the forcing inputs are not measured, and that only a subset of the nodes outputs is accessible. The developed technique is applicable to scenarios encompassing confounding variables and feedback loops, which are complicating factors potentially introducing bias in the estimate of the transfer function. The results are based on the prediction of the output node using the input node along with a set of additional auxiliary variables which are selected only from the observed nodes. Similar prediction error methods provide only sufficient conditions for the appropriate choice of auxiliary variables and assume a priori information about the location of strictly causal operators in the network. In this article, such an a priori knowledge is not required. A most remarkable feature of our approach is that the conditions for the selection of the auxiliary variables are purely graphical. Furthermore, within single-output prediction methods such conditions are proven to

The article revisits Aizerman and Kalman conjectures for absolute stability through the lens of a novel graphical interpretation. Even though these conjectures have been proven false, such a graphical interpretation suggests natural ways to introduce additional conditions in order to obtain new valid absolute stability criteria. As an illustration, the article proves a new absolute stability criterion obtained by the iterative application of a variation of the circle criterion.