Spectra of random networks with arbitrary degrees

Area of Science:

• Network science
• Statistical physics
• Graph theory

Background:

• Understanding the spectral properties of complex networks is crucial for analyzing their behavior.
• Existing methods often struggle with networks having arbitrary node degrees, such as the configuration model.

Purpose of the Study:

• To derive a message-passing method for computing spectra of locally treelike networks.
• To develop an approximation for computing the spectral density of random graphs (configuration model).
• To establish bounds on spectral band edges for identifying network phase transitions.

Main Methods:

• Derivation of a message-passing algorithm for network spectra computation.
• Development of an approximation for the spectral density of the configuration model.
• Derivation of bounds on the spectrum's band edges.

Main Results:

• The message-passing method provides accurate spectral computations for locally treelike networks.
• The approximation yields accurate spectral density for the configuration model in all but the sparsest networks.
• Derived bounds on band edges are useful for detecting structural phase transitions.

Conclusions:

• The developed message-passing method and its approximation offer efficient tools for analyzing the spectra of complex networks.
• The findings are particularly relevant for understanding random graphs and identifying critical network properties.
• The work provides insights into network structure and phase transitions through spectral analysis.