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Properties of dense partially random graphs.

Sebastián Risau-Gusman¹

¹Instituto de Física, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre, RS, Brazil. srisau@if.ufrgs.br

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 17, 2004

Summary

We analyzed random graphs with defined neighborhoods, finding properties similar to small-world graphs (SWG). Our study reveals eigenvalue distributions and their impact on graph mixing and synchronization.

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Area of Science:

Graph Theory
Network Science
Statistical Physics

Background:

Traditional random graphs lack local structure.
Small-world graphs (SWGs) exhibit high clustering and short path lengths.
Existing models often focus on sparse graphs.

Purpose of the Study:

Investigate random graphs with pre-defined neighborhoods and high average degrees.
Analyze graph properties like mean distance and clustering.
Characterize the eigenvalue distribution of adjacency matrices.

Main Methods:

Defining edge probability based on vertex neighborhood relationships.
Analytical calculation of mean distance and clustering.
Deriving the distribution of eigenvalues for adjacency matrices.

Main Results:

Properties qualitatively similar to SWGs, including mean distance and clustering.
Eigenvalue distribution shows a discrete part (rescaled substrate spectrum) and a continuous part.
The continuous part follows a semicircle law, with width indicating graph disorder.

Conclusions:

The studied random graph model offers insights into network structure and dynamics.
Results are applicable to calculating mixing rates and synchronizability thresholds.
The findings bridge properties of regular lattices and random graphs.

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