  This article addresses the problem of reconstructing the topology of a
network of agents interacting via linear dynamics, while being excited by
exogenous stochastic sources that are possibly correlated across the agents,
from time-series measurements alone. It is shown, under the assumption that the
correlations are affine in nature, such network of nodal interactions is
equivalent to a network with added agents, represented by nodes that are
latent, where no corresponding time-series measurements are available; however,
here all exogenous excitements are spatially (that is, across agents)
uncorrelated. Generalizing affine correlations, it is shown that, under
polynomial correlations, the latent nodes in the expanded networks can be
excited by clusters of noise sources, where the clusters are uncorrelated with
each other. The clusters can be replaced with a single noise source if the
latent nodes are allowed to have non-linear interactions. Finally, using the
sparse plus low-rank matrix decomposition of the imaginary part of the inverse
power spectral density matrix (IPSDM) of the time-series data, the topology of
the network is reconstructed. Under non conservative assumptions, the
correlation graph is retrieved.
