What is it about?

Spectral network identification aims at inferring the eigenvalues of the Laplacian matrix of a network from measurement data. This allows to capture global information on the network structure from local measurements at a few number of nodes. In this paper, we consider the spectral network identification problem in the generalized setting of a vector-valued diffusive coupling. The feasibility of this problem is investigated and theoretical results on the properties of the associated generalized eigenvalue problem are obtained. Finally, we propose a numerical method to solve the generalized network identification problem, which relies on dynamic mode decomposition and leverages the above theoretical results.

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Why is it important?

In some situations, only local measurements of a few nodes of a network are available, while the whole network topology is unknown. Since the network structure affects the dynamics of coupled units attached to the nodes, it is possible to infer some information on the global structure of the network from these local measurements. In particular, this can be done in the context of spectral network identification introduced in Mauroy and Hendrickx (2017). In this setting, the eigenvalues of the Jacobian matrix characterizing the linearized dynamics are computed from local measurement through the so-called DMD algorithm (see Schmid (2010); Tu et al. (2014)) and are shown to be related to the eigenvalues of the Laplacian matrix of the network. Thanks to spectral graph theory, the obtained Laplacian spectrum can in turn be used to infer global structural properties of the network such as the minimum, mean and maximum node degrees (see e.g. Fiedler (1973)). However, the results of Mauroy and Hendrickx (2017) are valid only for units interacting through a scalar diffusive coupling. This restrictive condition is not satisfied in many cases, such as reaction-diffusion networks. This paper aims to extend the applicability of the spectral network identification framework to the setting of vector-valued diffusive coupling between interacting units. In this context, the feasibility problem appears to be more involved than in the scalar-valued coupling case and theoretical results related to a generalized eigenvalue problem are provided. Furthermore, a method for spectral network identification is developed in this generalized case.

Perspectives

This work leads to several research perspectives. First of all, the proposed method is not robust to numerical errors on the eigenvalues estimated by the DMD algorithm. We have proposed a heuristic solution to this issue, which could be improved. In this context, specific extensions of the DMD algorithm could also be used (e.g., Brunton et al. (2013), Williams et al. (2015), Hemati et al. (2017), Li et al. (2017), Gulina and Mauroy (2021)). Moreover, the results and methods developed in this work are based on the assumption of a network of identical units. Further research work could investigate the case of nonidentical units with possibly several equilibria or even admitting a limit-cycle. Finally, additional work could be needed to infer relevant network properties from identified features of the Laplacian spectrum. While the spectra of the adjacency and modularity matrices of random graphs have been studied in Nadakuditi and Newman (2013), the reverse problem of estimating distributions of expected degree from the identified Laplacian spectrum could be addressed.

Dr Marvyn Vincenzo Gulina
Universite de Namur

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This page is a summary of: Spectral identification of networks with generalized diffusive coupling, IFAC-PapersOnLine, January 2022, Elsevier,
DOI: 10.1016/j.ifacol.2022.11.101.
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