Network susceptibilities: Theory and applications

Area of Science:

• Complex Systems Science
• Network Science
• Dynamical Systems Theory

Background:

• Understanding how complex networks respond to perturbations is crucial for their stability and function.
• Existing methods often lack a unified framework to quantify collective dynamic responses to localized changes.

Purpose of the Study:

• To introduce and define the concept of network susceptibilities.
• To differentiate between vertex and edge susceptibilities for analyzing network responses.
• To provide a generalized framework applicable to diverse network types.

Main Methods:

• Derivation of explicit formulas for network susceptibilities in oscillator networks near steady states.
• Application of the framework to Kuramoto-type phase-oscillator models.
• Analysis of power grid models and generic flow models.

Main Results:

• Defined vertex susceptibilities (response to unit property changes) and edge susceptibilities (response to interaction changes).
• Derived analytical expressions for these susceptibilities in specific network models.
• Demonstrated the applicability to oscillator, power grid, and flow networks.

Conclusions:

• Network susceptibilities offer a powerful quantitative tool for analyzing the collective dynamics of complex systems.
• The framework is generalizable to various network types, particularly those involving flow, transport, or spreading phenomena.
• This concept is essential for understanding network responses to topological and parameter changes.