Isolation and Connectivity in Random Geometric Graphs with Self-similar Intensity Measures

Area of Science:

• Graph theory
• Network science
• Probability theory

Background:

• Random geometric graphs (RGGs) model networks where nodes are points in space.
• Connectivity in RGGs is often analyzed using uniform node distributions (e.g., on a square).
• The Poisson distribution of isolated nodes and its relation to graph connectivity are key properties in the standard RGG model.

Purpose of the Study:

• To investigate the properties of isolation and connectivity in RGGs with various self-similar node distributions.
• To analyze the impact of nonuniform, fractal, and finitely ramified node distributions on graph properties.
• To compare these properties with the standard uniform distribution model.

Main Methods:

• Analysis of self-similar node distributions, including smooth, fractal, uniform, and nonuniform types.
• Mathematical examination of node isolation and graph connectivity under different distribution models.
• Evaluation of integral calculations and analytical arguments for both smooth and fractal distributions.

Main Results:

• Nonuniform node distributions can disrupt the Poisson distribution of isolated nodes.
• Nonuniformity enhances the correlation between isolated nodes and overall graph connectivity.
• Connectivity transitions are broadened by nonuniform distributions, and finite ramification introduces further connectivity challenges.
• Fractal distributions exhibit similar behaviors to smooth distributions, with some analytical differences.

Conclusions:

• The choice of node distribution significantly influences the connectivity properties of random geometric graphs.
• Nonuniformity and fractal structures present distinct challenges and behaviors compared to standard uniform models.
• Understanding these variations is crucial for accurately modeling real-world networks with complex spatial distributions.