Exact topology identification of large-scale interconnected dynamical systems from compressive observations

In this paper, we consider the problem of identifying the exact topology of an interconnected dynamical network from a limited number of measurements of the individual nodes. Within the network graph, we assume that interconnected nodes are coupled by a discrete-time convolution process, and we explain how, given observations of the node outputs, the problem of topology identification can be cast as solving a linear inverse problem. We use the term compressive observations in the case when there is a limited number of measurements available and thus the resulting inverse problem is highly underdetermined. Inspired by the emerging field of Compressive Sensing (CS), we then show that in cases where network interconnections are suitably sparse (i.e., the network contains sufficiently few links), it is possible to perfectly identify the topology from small numbers of node observations, even though this leaves a highly underdetermined set of linear equations. This can dramatically reduce the burden of data acquisition for problems involving network identification. The main technical novelty of our approach is in casting the identification problem as the recovery of a block-sparse signal x ∈ R^N from the measurements b = Ax ∈ R^M with M <; N, where the measurement matrix A is a block-concatenation of Toeplitz matrices. We discuss identification guarantees, introduce the notion of network coherence for the analysis of interconnected networks, and support our discussions with illustrative simulations.