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Spectra and eigenvectors of scale-free networks
1School of Physics, Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 12, 2001
Summary
We analyzed the spectral properties of scale-free networks with bidirectional interactions. The largest eigenvalue scales with system size, and its corresponding eigenfunction localizes at the network hub.
Area of Science:
- Network Science
- Spectral Graph Theory
- Complex Systems
Background:
- Scale-free networks are crucial models for complex systems.
- Understanding their spectral properties reveals network dynamics.
- Bidirectional interactions create real, symmetric adjacency matrices.
Purpose of the Study:
- To investigate the spectra and eigenvectors of scale-free networks with bidirectional interactions.
- To characterize the behavior of eigenvalues and eigenfunctions, particularly the largest ones.
- To determine scaling laws for spectral properties with system size.
Main Methods:
- Analysis of adjacency matrices for scale-free networks.
- Calculation and examination of eigenvalues and eigenvectors.
- Investigation of spectral density and eigenfunction localization.
- Determination of scaling relationships with system size (N).
Main Results:
- Spectral density exhibits exponential decay and power-law tails.
- The largest eigenvalue (λ1) scales as approximately N^(1/4).
- The corresponding eigenfunction localizes at the network hub (highest degree vertex).
- The mass gap scales as N^(-0.68).
Conclusions:
- The spectral properties of these networks are distinct, with localized eigenfunctions at hubs.
- Scaling laws provide insights into network behavior and stability.
- Findings are relevant for understanding information flow and robustness in complex networks.
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