Open networks in discrete time: Passing vs blocking behavior

Area of Science:

• Network Science
• Systems Theory
• Control Theory

Background:

• Complex networks are often modeled in discrete time, including opinion dynamics, Markov chains, and diffusion processes.
• Understanding how these networks interact with their environment as open systems is crucial for analyzing information flow.
• Existing methods may not efficiently characterize the input-output behavior of discrete-time complex networks.

Purpose of the Study:

• To develop a unified framework for analyzing the input-output behavior of discrete-time complex networks as open systems.
• To characterize whether these networks amplify (pass) or suppress (block) external inputs.
• To provide a computationally efficient method for comparing network topologies based on their signal processing capabilities.

Main Methods:

• Combining the network's transfer function with the discrete-time controllability Gramian.
• Utilizing the H2-norm to measure signal gain across diverse input types.
• Introducing a network index based on Gramian trace and eigenvalues for scalable analysis.

Main Results:

• A general framework is established for characterizing signal amplification or suppression in discrete-time networks.
• A computationally efficient network index is developed, enabling scalable comparisons of network topologies.
• Empirical networks across biological, technological, and ecological domains exhibit consistent structural signatures related to passing or blocking behavior.

Conclusions:

• The developed framework effectively analyzes the input-output dynamics of discrete-time complex networks.
• Network architecture and the choice of input/output nodes significantly shape information flow.
• Findings have broad implications for network control, signal processing, and the design of complex systems.