Open networks in discrete time: Passing vs blocking behavior

This paper presents a unified framework for analyzing the input-output behavior of discrete-time complex networks viewed as open systems. Importantly, we focus on systems that are inherently modeled in discrete time-such as opinion dynamics, Markov chains, diffusion on networks, and population models-reflecting their natural formulation in many real-world contexts. By an open network, we mean one that is coupled to its environment, through both external signals that are received by designated input nodes and response signals that are released back into the environment via a separate set of output nodes. We develop a general framework for characterizing whether such networks amplify (pass) or suppress (block) the external inputs. Our approach combines the transfer function of the network with the discrete-time controllability Gramian, and uses the H2-norm as a comprehensive measure of signal gain across various classes of inputs. We introduce a computationally efficient network index based on the Gramian trace and eigenvalues, enabling scalable comparisons across network topologies. Application of our method to a broad set of empirical networks-spanning biological, technological, and ecological domains-uncovers consistent structural signatures associated with passing or blocking behavior. These findings shed light on how the network architecture and the particular selection of input and output nodes shape information flow in real-world systems, with broad implications for control, signal processing, and network design.