Random hyperbolic graphs in $d+1$ dimensions

Abstract

We consider random hyperbolic graphs in hyperbolic spaces of any dimension $d+1\ge 2$. We present a rescaling of model parameters that casts the random hyperbolic graph model of any dimension to a unified mathematical framework, leaving the degree distribution invariant with respect to the dimension. Unlike the degree distribution, clustering does depend on the dimension, decreasing to 0 at $d\to \infty$. We analyze all of the other limiting regimes of the model, and we release a software package that generates random hyperbolic graphs and their limits in hyperbolic spaces of any dimension.

References in corpus (27)

• Hyperbolic Geometry of Complex Networks
• Popularity versus Similarity in Growing Networks
• Navigability of Complex Networks
• Sustaining the Internet with Hyperbolic Mapping
• Class of correlated random networks with hidden variables
• Self-similarity of complex networks and hidden metric spaces
• Network Geometry
• Machine learning meets network science: dimensionality reduction for fast and efficient embedding of networks in the hyperbolic space
• Scale-free Networks Well Done
• Network Mapping by Replaying Hyperbolic Growth
• Curvature and temperature of complex networks
• Multiscale unfolding of real networks by geometric renormalization
• Emergence of Soft Communities from Geometric Preferential Attachment
• Network Geometry Inference using Common Neighbors
• Navigable Networks as Nash Equilibria of Navigation Games
• Principled approach to the selection of the embedding dimension of networks
• Hyperbolic Graph Generator
• Link prediction with hyperbolic geometry
• Small worlds and clustering in spatial networks
• Detecting the ultra low dimensionality of real networks
• Navigability evaluation of complex networks by greedy routing efficiency
• Geohyperbolic Routing and Addressing Schemes
• Navigating ultrasmall worlds in ultrashort time
• Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
• Finding shortest and nearly shortest path nodes in large substantially incomplete networks
• A geometry-induced topological phase transition in random graphs
• Growing hyperbolic networks beyond two dimensions: the generalised popularity-similarity optimisation model

Cited by in corpus (6)

• Detecting the ultra low dimensionality of real networks
• Growing hyperbolic networks beyond two dimensions: the generalised popularity-similarity optimisation model
• Greedy routing optimisation in hyperbolic networks
• Community detection robustness of graph neural networks
• The Shortest-Path distance on graphons
• $({\omega }_1,{\omega }_2)$-temporal random hyperbolic graphs