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Spectral properties of directed random networks with modular structure
Sarika Jalan1, Guimei Zhu, Baowen Li
1School of Sciences, Indian Institute of Technology Indore, IET-DAVV Campus, Khandwa Road, Indore 452017, India. sarikajalan9@gmail.com
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 21, 2011
Summary
We found that correlated entries in directed networks lead to localized eigenvectors, unlike uncorrelated networks with delocalized ones. Network structure, like community organization, significantly impacts this localization behavior.
Area of Science:
- Complex systems
- Network science
- Theoretical physics
Background:
- Understanding the spectral properties of directed networks is crucial for various fields, including neuroscience and statistical physics.
- Eigenvector localization in random networks is influenced by the correlation structure of network matrices.
Purpose of the Study:
- To investigate the impact of correlations and network structure on eigenvector localization in directed networks.
- To explore the transition from delocalized to localized eigenvectors as network properties change.
- To analyze the spectral properties of a real-world metabolic network and compare it to model systems.
Main Methods:
- Analysis of network spectra and eigenvector localization measures.
- Systematic variation of correlation strengths, connection probability, and directionality in model networks.
- Examination of network community structure and its effect on spectral properties.
- Comparison of model network results with spectral analysis of a zebrafish metabolic network.
Main Results:
- Uncorrelated random networks exhibit circular eigenvalue distributions with delocalized eigenvectors.
- Correlated network entries lead to localized eigenvectors.
- Directed connections introduce complex-conjugate pairs in eigenstates, creating rich spectral patterns.
- Strong community structure results in localized spectra, with deviations causing abrupt localization changes.
- Zebrafish metabolic network spectral properties show similarities to model networks.
Conclusions:
- Network correlations and community structure are key determinants of eigenvector localization.
- The transition to localization is sensitive to small structural perturbations in networks with community structure.
- Spectral analysis provides insights into the organization and dynamics of complex systems, including biological networks.
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