Spectral identification of networks with inputs

We consider a network of interconnected dynamical systems. Spectral network identification consists in recovering the eigenvalues of the network Laplacian from the measurements of a very limited number (possibly one) of signals. These eigenvalues allow to deduce some global properties of the network, such as bounds on the node degree. Having recently introduced this approach for autonomous networks of nonlinear systems, we extend it here to treat networked systems with external inputs on the nodes, in the case of linear dynamics. This is more natural in several applications, and removes the need to sometimes use several independent trajectories. We illustrate our framework with several examples, where we estimate the mean, minimum, and maximum node degree in the network. Inferring some information on the leading Laplacian eigenvectors, we also use our framework in the context of network clustering.

[1]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[2]  M. Timme,et al.  Revealing networks from dynamics: an introduction , 2014, 1408.2963.

[3]  W. Zheng,et al.  Optimization-based structure identification of dynamical networks , 2013, Physica A: Statistical Mechanics and its Applications.

[4]  Xavier Bombois,et al.  Identification of Dynamic Models in Complex Networks With Prediction Error Methods: Predictor Input Selection , 2016, IEEE Transactions on Automatic Control.

[5]  Andrzej Banaszuk,et al.  Hearing the clusters of a graph: A distributed algorithm , 2009, Autom..

[6]  Ulrich Parlitz,et al.  Driving a network to steady states reveals its cooperative architecture , 2008 .

[7]  Andrea Gasparri,et al.  Decentralized estimation of Laplacian eigenvalues in multi-agent systems , 2012, Autom..

[8]  T. Sauer,et al.  Reconstructing the topology of sparsely connected dynamical networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Wen-Xu Wang,et al.  Noise bridges dynamical correlation and topology in coupled oscillator networks. , 2010, Physical review letters.

[10]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[11]  Julien M. Hendrickx,et al.  Spectral Identification of Networks Using Sparse Measurements , 2016, SIAM J. Appl. Dyn. Syst..

[12]  Takashi Hikihara,et al.  Data-Driven Partitioning of Power Networks Via Koopman Mode Analysis , 2016, IEEE Transactions on Power Systems.

[13]  Michel Gevers,et al.  On the identifiability of dynamical networks , 2017 .

[14]  Marc Timme,et al.  Inferring network topology from complex dynamics , 2010, 1007.1640.

[15]  Victor M. Preciado,et al.  Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks , 2011, IEEE Transactions on Automatic Control.

[16]  Alain Y. Kibangou,et al.  Distributed estimation of Laplacian eigenvalues via constrained consensus optimization problems , 2015, Syst. Control. Lett..

[17]  Wen-Xu Wang,et al.  Time-series–based prediction of complex oscillator networks via compressive sensing , 2011 .

[18]  Ljupco Kocarev,et al.  Estimating topology of networks. , 2006, Physical review letters.

[19]  Steven L. Brunton,et al.  On dynamic mode decomposition: Theory and applications , 2013, 1312.0041.

[20]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[21]  Steven L. Brunton,et al.  Dynamic Mode Decomposition with Control , 2014, SIAM J. Appl. Dyn. Syst..