Spectral identification of networks using sparse measurements

TL;DR

This paper introduces a spectral network identification method that infers global network properties from sparse measurements at a few nodes, leveraging spectral graph theory and Koopman operator analysis.

Contribution

It develops a novel framework connecting spectral properties of networks to dynamical data, enabling global inference from minimal local observations.

Findings

Can detect the mean number of connections from a single node

Identifies addition of new vertices using sparse measurements

Applicable to large, heterogeneous networks

Abstract

We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification. The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the Dynamic Mode Decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations. Our framework provides efficient numerical methods to infer global information on the network from sparse local measurements at a few nodes. Numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node, that need not be representative of the other nodes' properties.