  We study the evolution of a random weighted network with complex nonlinear
dynamics at each node, whose activity may cease as a result of interactions
with other nodes. Starting from a knowledge of the micro-level behaviour at
each node, we develop a macroscopic description of the system in terms of the
statistical features of the subnetwork of active nodes. We find the asymptotic
characteristics of this subnetwork to be remarkably robust: the size of the
active set is independent of the total number of nodes in the network, and the
average degree of the active nodes is independent of both the network size and
its connectivity. These results suggest that very different networks evolve to
active subnetworks with the same characteristic features. This has strong
implications for dynamical networks observed in the natural world, notably the
existence of a characteristic range of links per species across ecological
systems.
