  We consider performance deterioration of interconnected linear dynamical
networks subject to exogenous stochastic disturbances. The focus of this paper
is on first-order and second-order linear consensus networks. We employ the
expected value of the steady state dispersion of the state of the entire
network as a performance measure and develop a graph-theoretic methodology to
relate structural specifications of the underlying graphs of the network to the
performance measure. We explicitly quantify several inherent fundamental limits
on the best achievable levels of performance in linear consensus networks and
show that these limits of performance are merely imposed by the specific
structure of the underlying graphs. Furthermore, we discover new connections
between notions of sparsity and the performance measure. Particularly, we
characterize several fundamental tradeoffs that reveal interplay between the
performance measure and various sparsity measures of a linear consensus
network. At the end, we apply our results to two real-world dynamical networks
and provide energy interpretations for the proposed performance measures. It is
shown that the total power loss in synchronous power networks and total kinetic
energy of a network of autonomous vehicles in a formation are viable
performance measure for these networks and fundamental limits on these measures
quantify the best achievable levels of energy-efficiency in these dynamical
networks.
