Robust emergent activity in dynamical networks.

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 microlevel behavior 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 that very different networks evolve to active subnetworks with similar asymptotic characteristics: 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. This robustness 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.