  Positive Lyapunov exponents measure the asymptotic exponential divergence of
nearby trajectories of a dynamical system. Not only they quantify how chaotic a
dynamical system is, but since their sum is an upper bound for the entropy by
the Ruelle inequality, they also provide a convenient way to quantify the
complexity of an active network. We present numerical evidences that for a
large class of active networks, the sum of the positive Lyapunov exponents is
bounded by the sum of the positive Lyapunov exponents of the corresponding
synchronization manifold, the last quantity being in principle easier to
compute than the latter. This fact is a consequence of the property that for an
active network considered here, the amount of information produced is more
affected by the interactions between the nodes than by the topology of the
network. Using the inequality described above, we explain how to predict the
behavior of a large active network only knowing the information provided by an
active network consisting of two coupled nodes.
