  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.
