Open networks in discrete time: Passing vs blocking behavior

TL;DR

The paper tackles how discrete-time open networks propagate external signals and how topology and input/output placement determine amplification or attenuation. It presents a unified approach built on the discrete-time $\mathcal{H}_2$-norm and the output controllability Gramian $W_d^{out}$, plus a simple distance-$d$ based approximation $\hat{W}_d^{out} \approx \dfrac{(C A^d B)^2}{1-\rho^2}$. A computationally cheap network index $\alpha$ is proposed to quantify input placement effects, and extensive empirical validation across biological, technological, and ecological networks shows systematic passing vs blocking tendencies. Together these results enable scalable assessment and design of information flow in complex networks, with implications for control, signal processing, and robust network design, and connect to broader work on non-normal directed networks.

Abstract

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, using the H2-norm to quantify signal amplification. 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.

Figures (6)

• Figure 1: Randomly generated graph for Example 1, where the adjacency matrix $A$, the input matrix $B$, and the output matrix $C$ are provided on the right side of the figure. Input nodes are represented in blue color and output nodes in red color.
• Figure 2: Panels a-c show a comparison between the true (Eq. \ref{['eq:wout']}) and approximated (Eq. \ref{['eq:Wapprox']}) output controllability Gramian. The blue color (red color) is for the case that the distance from the input to the output node is $d=1$ ($d=2$). Circles are true values and stars are approximated values. We see that the output controllability Gramian $W_d^{out}$ decays for larger ER networks (larger $N$), for more dense networks (larger $p$), and for longer distances between the input and the output nodes (larger $d$). The figure only shows the two cases that the distance from the input node to the output node is either $d=1$ or $d=2$, with the latter yielding a much better performance of the approximation than the former (see also the lower panels). Larger values of $d$, for which the approximation works even better are not shown. Panels d-f show the normalized error $E$ (Eq. \ref{['eq:normerr']}) as the connection probability $p$ of Erdős-Rényi graph is varied, for different numbers of nodes $N$. We see that the error $E$ decays with the distance $d$. The data is averaged over 100 realizations for each choice of $(d,p)$ in all panels.
• Figure 3: Panels a-c show a comparison between the true (Eq. \ref{['eq:wout']}) and approximated (Eq. \ref{['eq:Wapprox']}) output controllability Gramians for different values of the path length $d$ from the input to the output node, as the control parameter $\beta$ of the scale-free network is varied, for different numbers of the network nodes $N$. The degree distribution of the scale free networks is more homogeneous for larger values of $\beta$. For example, the choice $\beta=0.5$ ($\beta=1$) corresponds to a more homogeneous (more heterogeneous) scale-free network with $\gamma=3$ ($\gamma=2$). Circles are true values and stars are approximated values. Panels d-f show the normalized error $E$ (Eq. \ref{['eq:normerr']}) as the control parameter $\beta$ of the scale-free network is varied, for different numbers of nodes $N$. In all panels we see that the error decreases with the distance $d$. The data is averaged over 100 realizations for each choice of $(d,\beta)$ in all panels.
• Figure 4: The trace of the infinite-horizon discrete-time controllability Gramian $\text{Tr}(W_d)$ for different real networks vs their number of input nodes $M$. In panels a and b, the adjacency matrices are scaled such that all networks have spectral radii of 0.1 and 0.98, respectively. The dashed black line represents $\text{Tr}(W_d) = M$. We see that when the spectral radii of the adjacency matrices are small (panel a), $\text{Tr}(W_d) \approx M$, and when the spectral radii are large (panel b), $\text{Tr}(W_d) \geq M$.
• Figure 5: The trace of the infinite-horizon discrete-time controllability Gramian $\text{Tr}(W_d)$ vs the network index $\alpha$ for selected real network datasets (IEEE118, UK Grid, Cat, C-Elegans, Mouse Liver, Mouse Brain, Stem Cells 44, and Macaque 47). For all datasets, the spectral radius of the adjacency matrix is scaled to be $\rho=0.2$. Each dataset has its own unique number of input nodes $M$. We randomly choose 100 sets of $M$ input nodes and evaluate $\text{Tr}(W_d)$ and $\alpha$ for each set (blue circles). The pair value of $\text{Tr}(W_d)$ and $\alpha$ from the input nodes within each real dataset is plotted as a red diamond. In each panel, a green star is used to label the optimal selection of $M$ nodes, as described in the text. Due to the large size of the datasets C-Elegans and Mouse Brain, we were not able to perform the optimization. In all panels, we see an approximately linear relationship between the network index $\alpha$ and the trace of the (output) controllability Gramian, which is equal to the $\mathcal{H}_2$-norm squared.

Theorems & Definitions (1)

• Remark 1