The frequency response of networks as open systems

Many biological, technological, and social systems can be effectively described as networks of interacting subsystems. Typically, these networks are not isolated objects, but interact with their environment through both signals and information that are received by specific nodes with an input function or released to the environment by other nodes with an output function. An important question is whether the structure of different networks, together with the particular selection of input and output nodes, is such that it favors the passing or blocking of such signals. For a given network and a given choice of the input and output nodes, the $H_2$ -norm provides a natural and general quantification of the extent to which input signals-whether deterministic or stochastic, periodic or arbitrary-are amplified. We analyze a diverse set of empirical networks and find that many naturally occurring systems, such as food webs, signaling pathways, and gene regulatory circuits, are structurally organized to enhance the passing of signals; in contrast, the structure of engineered systems like power grids appears to be intentionally designed to suppress signal propagation.