Characterizing network directedness using the containing pseudospectral frontier of the network Laplacian.
Christopher J Keylock1, Maurizio Carbone2
1Loughborough University, School of Architecture, Building and Civil Engineering, Loughborough, United Kingdom.
Physical Review. E
|April 18, 2026
Summary
This study introduces a novel method to quantify network directedness using the pseudospectrum of the Laplacian. The approach reveals insights into complex systems and aids in validating turbulence models.
Area of Science:
- Network science
- Applied mathematics
- Fluid dynamics
Background:
- Complex systems are often modeled as directed networks.
- Existing network analysis methods primarily focus on undirected graphs.
- Characterizing directedness is crucial for understanding system dynamics.
Purpose of the Study:
- To propose a novel method for quantifying the relative directedness of complex networks.
- To extend existing definitions of network directedness.
- To apply the method to analyze the dynamics of turbulent flow.
Main Methods:
- Utilizing the pseudospectrum of the network Laplacian.
- Comparing the pseudospectral frontier of the Laplacian with a variant lacking non-normal structure.
- Developing a metric based on the distance between these frontiers to characterize directedness.
- Applying the metric to a network representing Lagrangian dynamics in turbulent flow.
Main Results:
- The proposed metric effectively characterizes network directedness, consistent with theoretical expectations.
- Application to turbulent flow dynamics revealed two local maxima in the complex plane.
- Analysis highlighted specific atypical nodes influencing these maxima, differing from those identified by a stochastic differential equation model.
Conclusions:
- The pseudospectral approach provides a sensitive metric for network directedness.
- This method offers potential for network characterization and model validation, particularly in complex systems like turbulent flows.
- The findings underscore the importance of considering network structure in analyzing system dynamics.
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