  What does the informational complexity of dynamical networked systems tell us
about intrinsic mechanisms and functions of these complex systems? Recent
complexity measures such as integrated information have sought to
operationalize this problem taking a whole-versus-parts perspective, wherein
one explicitly computes the amount of information generated by a network as a
whole over and above that generated by the sum of its parts during state
transitions. While several numerical schemes for estimating network integrated
information exist, it is instructive to pursue an analytic approach that
computes integrated information as a function of network weights. Our
formulation of integrated information uses a Kullback-Leibler divergence
between the multi-variate distribution on the set of network states versus the
corresponding factorized distribution over its parts. Implementing stochastic
Gaussian dynamics, we perform computations for several prototypical network
topologies. Our findings show increased informational complexity near
criticality, which remains consistent across network topologies. Spectral
decomposition of the system's dynamics reveals how informational complexity is
governed by eigenmodes of both, the network's covariance and adjacency
matrices. We find that as the dynamics of the system approach criticality, high
integrated information is exclusively driven by the eigenmode corresponding to
the leading eigenvalue of the covariance matrix, while sub-leading modes get
suppressed. The implication of this result is that it might be favorable for
complex dynamical networked systems such as the human brain or communication
systems to operate near criticality so that efficient information integration
might be achieved.
