Spectral analysis and the dynamic response of complex networks

Area of Science:

• Network Science
• Complex Systems Analysis
• Computational Biology

Background:

• Eigenvalues and eigenvectors of connectivity matrices encode network topology and behavior.
• Spectral density distributions (rho(lambda)) differentiate random (Wigner's law) and scale-free (triangular) networks.

Purpose of the Study:

• To characterize the spectral density of hierarchical networks.
• To identify a unique spectral fingerprint for modularity in complex networks.
• To evaluate the adequacy of current network models for biological systems.

Main Methods:

• Analysis of the spectral density (rho(lambda)) of network connectivity matrices.
• Comparison of spectral density patterns across random, scale-free, and hierarchical network models.
• Examination of rho(0) as an indicator of network homeostatic response.

Main Results:

• Hierarchical networks exhibit a distinct spectral density pattern, serving as a fingerprint for modularity.
• The value rho(0) is significantly smaller in hierarchical modular networks compared to random or scale-free networks.
• Biological protein-protein interaction networks show a large rho(0), indicating current models are inadequate.

Conclusions:

• The spectral density of hierarchical networks provides a novel method for identifying modularity.
• Hierarchical modular networks possess distinct topological and homeostatic properties.
• Existing network models fail to accurately represent the structure of biological protein-protein interaction networks.