The Frequency Response of Networks as Open Systems

Many biological, technological, and social systems are effectively networks of interacting individual systems. Typically, these networks are not isolated objects, but interact with their environment through both signals and information that is 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 H2-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 conjecture that many naturally occurring systems-such as food webs, signaling pathways, and gene regulatory circuits-are structurally organized to enhance the passing of signals, facilitating the efficient flow of biomass, information, or regulatory activity. This passing behavior culminates in directed acyclic graphs (DAGs), for which we analytically show that amplification depends on the number and length of input-output pathways, which is consistent with the well-known tendency of naturally emerging networks to approximate DAG structures. In contrast, the structure of engineered systems like power grids appears to be intentionally designed to suppress signal propagation, as the transmitted quantity-voltage phase differences-requires tight control to maintain synchronized operation.