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Random graphs containing arbitrary distributions of subgraphs.

Brian Karrer¹, M E J Newman

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA.

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

|January 15, 2011

Summary

This study introduces new random graph models that realistically capture network structures with loops and cliques, moving beyond unrealistic treelike models. These models allow for the analysis of fundamental network properties, including component size and phase transitions.

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Area of Science:

Network Science
Graph Theory
Statistical Physics

Background:

Traditional random graph models generate unrealistic treelike network neighborhoods.
Real-world networks often exhibit complex structures like loops and cliques.

Purpose of the Study:

To propose and analyze novel random graph models incorporating general subgraphs.
To enable the study of networks with realistic, non-treelike local structures.
To maintain analytical solvability for fundamental network properties.

Main Methods:

Development of a new class of random graph models.
Analysis of network properties within these models.
Application of percolation theory (site and bond).

Main Results:

The proposed models allow for non-treelike neighborhoods containing loops and cliques.
Solutions are provided for the size of the giant component.
The position of the phase transition for giant component emergence is determined.
Percolation properties for both site and bond percolation are analyzed.

Conclusions:

The new models offer a more realistic representation of real-world networks.
These models retain analytical tractability for key network characteristics.
The framework facilitates deeper understanding of network structure and behavior.