page_title: Hyperbolic geometric graph
section_title: Generating hyperbolic geometric graphs
hierarchical_section_title: Hyperbolic geometric graph / Generating hyperbolic geometric graphs
caption_attribution_description: English: The six graph were generated randomly and contain N=100 nodes each. Different values for alpha and the radius R of the hyperbolic disk were used. The graphs are drawn in the native representation.
context_page_description: A hyperbolic geometric graph or hyperbolic geometric network is a special type of spatial network where latent coordinates of nodes are sprinkled according to a probability density function into a
  hyperbolic space of constant negative curvature and an edge between two nodes is present if they are close according to a function of the metric. A HGG generalizes a random geometric graph whose embedding space is Euclidean.
context_section_description: Krioukov et al. describe how to generate hyperbolic geometric graphs with uniformly random node distribution (as well as generalized versions) on a disk of radius  in . These graphs yield a power-law distribution for the node degrees. The angular coordinate  of each point/node is chosen uniformly random from , while the density function for the radial coordinate r is chosen according to the probability distribution :
  The growth parameter  controls the distribution: For , the distribution is uniform in , for smaller values the nodes are distributed more towards the center of the disk and for bigger values more towards the border.  In this model, edges between nodes  and  exist iff  or with probability  if a more general connectivity decay function is used. The average degree is controlled by the radius  of the hyperbolic disk. It can be shown, that for  the node degrees follow a power law distribution with exponent .
  
  The image depicts randomly generated graphs for different values of  and  in . It can be seen how  has an effect on the distribution of the nodes and  on the connectivity of the graph. The native representation where the distance variables have their true hyperbolic values is used for the visualization of the graph, therefore edges are straight lines.
caption: Random hyperbolic geometric graphs with N=100 nodes each for different values of alpha and R
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