The interest on networks of dynamical systems is increasing in the past years, especially because of their capability of modeling and describing a large variety of phenomena and behaviors. Particular attention has been oriented towards the emergence of complicated phenomena from the interconnections of simple models. We tackle, from a theoretical perspective, the problem of reconstructing the topology of an unknown network of linear dynamical systems where every node output is given by a scalar random process. Existing approaches are bayesian or assume that the node inputs are accessible and manipulable. The approach we propose is completely blind, since we assume no a-priori knowledge of the network dynamics and we make use of second order statistics only. We also assume that it is only possible to observe the node outputs, while the inputs are not manipulable. The developed reconstruction technique is based on Wiener filtering and provides general theoretical guarantees for the detection of links in a network of dynamical systems. For a large class of networks, that we name "self-kin", sufficient conditions for the detection of the existing links are formulated. The necessity of those conditions is also discussed: indeed, it is shown that they do not hold only in specific pathological cases. Thus, for any applications needs, the exact reconstruction of self-kin networks can be considered practically guaranteed. For networks not belonging to this class those conditions are met by the smallest (in the sense of the least number of edges) self-kin network containing the actual one. Hence, for general networks, the procedure identifies a self-kin network that is optimal in the sense of number of edges. For networks not belonging to the self-kin class, conditions are met by the smallest (in the sense of the least number of edges) self-kin network containing the actual one.

Abstract-The paper deals with the problem of reconstructing the topological structure of a network of dynamical systems. A distance function is defined in order to evaluate the "closeness" of two processes and a few useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identification procedure for networked linear systems with tree topology are provided as well. Finally, the application of the techniques to the analysis of an actual complex network, i.e. to high frequency time series of the stock market, is illustrated.

No abstract

In this article we present a method to reconstruct the interconnectedness of dynamically related stochastic processes, where the interactions are bi-directional and the underlying topology is a tree. Our approach is based on multivariate Wiener filtering which recovers spurious edges apart from the true edges in the topology reconstruction. The main contribution of this work is to show that all spurious links obtained using Wiener filtering can be eliminated if the underlying topology is a tree based on which we present a three stage network reconstruction procedure for trees. We illustrate the effectiveness of the method developed by applying it on a typical distribution system of the electric grid. I. INTRODUCTIONNetworks underpin a powerful framework for modeling and analysis of large scale dynamical systems. Applications include neuroscience [1], financial markets [2], protein dynamics [3], climate sciences [4] and the power grid [5]. Moreover, networks play an indispensable role in building foundational aspects of control theory [6], statistical inference [7] and optimization theory [8]. The compactness of representation and the capability of unveiling influences, cause-effect relationships and dependencies amongst many variables are some of the key attributes enabled by network based approaches [9], [10]. An essential aspect of many studies is to determine a graphical representation of how multiple sub-systems/agents interact from measured time series data. It is often the case that active manipulation of the system is prohibited or not possible; for example, in financial markets the prices of stocks are available as data but it is not possible (or not allowed) to manipulate the prices. In many cases, the influences between sub-systems/ agents is mutual, thus separating source and destination or cause and effect in such cases is not meaningful.In this article, we are concerned with the task of unveiling the network topology that relates multiple linear dynamical systems from temporal data, where it is not possible to excite the system externally. Here, we assume that the underlying network is bi-directed, that is, the influences between agents is mutual and describing cause-effect relationships is not obvious. We further restrict the study to systems where the interaction flow is well characterized by a tree structure.