DeepAI AI Chat
Log In Sign Up

The Wigner's Semicircle Law of Weighted Random Networks

by   Yusuke Sakumoto, et al.
Tokyo Metropolitan University
Kwansei Gakuin University

The spectral graph theory provides an algebraical approach to investigate the characteristics of weighted networks using the eigenvalues and eigenvectors of a matrix (e.g., normalized Laplacian matrix) that represents the structure of the network. However, it is difficult for large-scale and complex networks (e.g., social network) to represent their structure as a matrix correctly. If there is a universality that the eigenvalues are independent of the detailed structure in large-scale and complex network, we can avoid the difficulty. In this paper, we clarify the Wigner's Semicircle Law for weighted networks as such a universality. The law indicates that the eigenvalues of the normalized Laplacian matrix for weighted networks can be calculated from the a few network statistics (the average degree, the average link weight, and the square average link weight) when the weighted networks satisfy the sufficient condition of the node degrees and the link weights.


page 1

page 2

page 3

page 4


Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

Network geometries are typically characterized by having a finite spectr...

Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model

We study the behavior of the eigenvectors associated with the smallest e...

Spectral bounds of the regularized normalized Laplacian for random geometric graphs

In this work, we study the spectrum of the regularized normalized Laplac...

Estimating the Spectral Density of Large Implicit Matrices

Many important problems are characterized by the eigenvalues of a large ...

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

We prove a central limit theorem for the components of the eigenvectors ...

Batch kernel SOM and related Laplacian methods for social network analysis

Large graphs are natural mathematical models for describing the structur...

Extended corona product as an exactly tractable model for weighted heterogeneous networks

Various graph products and operations have been widely used to construct...