On the problem of reconstructing an unknown topology

• Main Authors: Materassi, D, Salapaka, M V
• Format: Conference Proceeding
• Language: English
• Subjects: (t + τ )]: cross-covariance function; (w) = φ; (z) := φ; (z): power spectral density; (τ ) := E[X(t)Y; (τ ) := R; (τ )): cross-power spectral density; (τ ): autocovariance; 0; Algorithm design and analysis; A∣: cardinality (number of elements) of a set A; Biological system modeling; Biological systems; denotes the vector (x; E[{·]: mean operator; for the sake of simplicity; i) of a matrix W; Interconnected systems; is a vector of natural numbers; iw; Network topology; of stationary vector processes X and Y; Parallel algorithms; Power system modeling; R; Signal processing; Sufficient conditions; the conjugate transpose of a matrix or vector W; the entry (j; the i-th element of a vector x; The symbol := denotes a definition; the transpose of a matrix or vector W; when v = (i; Wiener filter; with abuse of notation; x; x∥: 2-norm of a vector x; Z(·): Zeta-transform of a signal; z) := Z(R
• ISSN: 0743-1619; 2378-5861
• DOI: 10.1109/ACC.2010.5531320

The interest for networks of dynamical systems has been 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 interconnections of simple models. In this paper we tackle, from a theoretical perspective, the problem of reconstructing the topology of an unknown network of dynamical systems. We propose a technique, based on Wiener filtering, which provides general theoretical guarantees for the detection of links in a network of dynamical systems. For a large class of network that we name "self-kin" sufficient conditions for a correct detection of a link are formulated. For networks not belonging to this class we give conditions for correct detection of links belonging to the smallest self-kin network containing the actual one.