Localizations on complex networks

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Complex networks lack Euclidean geometry, making traditional distance-based analysis difficult.
• Understanding network structure is crucial for fields ranging from biology to social sciences.

Purpose of the Study:

• To develop a method for characterizing the structural properties of complex networks.
• To investigate the localization phenomena within complex network structures.

Main Methods:

• Utilizing representative eigenvectors of the adjacency matrix to analyze network structure.
• Applying measures like participation ratio, structural entropy, and nearest neighbor level spacing distributions.
• Examining real-world whole-cell networks, Watts-Strogatz small-world networks, and Barabasi-Albert scale-free networks.

Main Results:

• The probability distribution function of eigenvector components effectively describes localization in non-Euclidean networks.
• Complex networks exhibit nontrivial localization properties stemming from their topological structures.
• The occurrence probabilities at nodes, when ranked, display general multifractal behavior.

Conclusions:

• Eigenvector component analysis provides a robust method for quantifying localization in complex networks.
• The observed multifractal characteristics serve as a valuable structural measure for complex networks.
• This approach enhances our understanding of the inherent structural organization of diverse complex systems.