Random hyperbolic graphs in d+1 dimensions

Area of Science:

• Graph Theory
• Hyperbolic Geometry
• Network Science

Background:

• Random hyperbolic graphs are a significant model in network science.
• Understanding their properties across different dimensions is crucial for theoretical and applied research.

Purpose of the Study:

• To develop a unified framework for random hyperbolic graphs in any dimension.
• To analyze the impact of dimensionality on graph properties like degree distribution and clustering.
• To provide a computational tool for generating and analyzing these graphs.

Main Methods:

• Introduced a rescaling of model parameters for a unified mathematical framework.
• Analyzed the degree distribution and clustering coefficients.
• Investigated other limiting regimes of the random hyperbolic graph model.
• Developed and released a software package.

Main Results:

• The degree distribution of random hyperbolic graphs is invariant to the dimension of the hyperbolic space.
• Clustering in these graphs is dimension-dependent, tending to zero as dimension increases (d→∞).
• All limiting regimes of the model were analyzed.

Conclusions:

• A dimension-independent framework for random hyperbolic graphs is established.
• Dimensionality significantly influences graph clustering but not degree distribution.
• The released software facilitates further research in this area.