Localizations on Complex Networks

Abstract

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.

9 pages, 6 fugures, 1 table

References in corpus (14)

• First-passage times in complex scale-invariant media
• Scaling theory of transport in complex networks
• Mutual information in random Boolean models of regulatory networks
• Exploring Complex Networks through Random Walks
• Geographical Coarsegraining of Complex Networks
• Localization Transition of Biased Random Walks on Random Networks
• Temporal Series Analysis Approach to Spectra of Complex Networks
• Heat conduction in simple networks: The effect of inter-chain coupling
• Entropy of complex relevant components of Boolean networks
• Von Neumann entropy and localization-delocalization transition of electron states in quantum small-world networks
• Quantum and classical diffusion in small-world networks
• Transport and Percolation Theory in Weighted Networks
• Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder
• Self-affine Fractals Embedded in Spectra of Complex Networks

Cited by in corpus (13)

• Quantitative features of multifractal subtleties in time series
• Random matrix analysis of localization properties of Gene co-expression network
• Universality in the spectral and eigenfunction properties of random networks
• Delocalization transition for the Google matrix
• Localization of eigenvectors in random graphs
• Spectral Properties of Directed Random Networks with Modular Structure
• Spectral statistics of random geometric graphs
• Scaling Properties of Multilayer Random Networks
• Normal mode analysis of spectra of random networks
• Scattering and transport properties of tight-binding random networks
• Geometrical and spectral study of $\beta$-skeleton graphs
• Localization in random bipartite graphs: numerical and empirical study
• Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix