Hyperbolic geometric graph
A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are sprinkled according to a probability density function into a hyperbolic space of constant negative curvature and (2) an edge between two nodes is present if they are close according to a function of the metric[1][2] (typically either a Heaviside step function resulting in deterministic connections between vertices closer than a certain threshold distance, or a decaying function of hyperbolic distance yielding the connection probability).
A HGG generalizes a random geometric graph (RGG) whose embedding space is Euclidean.
) and an edge set E constructed by considering the nodes as points placed onto a 2-dimensional hyperbolic space
, i.e. the radius of the Poincaré disk which can be visualized using a hyperboloid model.
The hyperbolic law of cosines allows to measure the distance
is established iff (if and only if) two nodes are within a certain neighborhood radius
In general, a link will be established with a probability depending on the distance
represents the probability of assigning an edge to a pair of nodes at distance
In this framework, the simple case of hard-code neighborhood like in random geometric graphs is referred to as truncation decay function.
[3] Krioukov et al.[2] describe how to generate hyperbolic geometric graphs with uniformly random node distribution (as well as generalized versions) on a disk of radius
These graphs yield a power-law distribution for the node degrees.
, while the density function for the radial coordinate r is chosen according to the probability distribution
the node degrees follow a power law distribution with exponent
The image depicts randomly generated graphs for different values of
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.
Source:[4] The naive algorithm for the generation of hyperbolic geometric graphs distributes the nodes on the hyperbolic disk by choosing the angular and radial coordinates of each point are sampled randomly.
For every pair of nodes an edge is then inserted with the probability of the value of the connectivity decay function of their respective distance.
is the number of nodes to generate, the distribution of the radial coordinate by the probability density function
denotes the uniform sampling of a value in the given interval.
Because the algorithm checks for edges for all pairs of nodes, the runtime is quadratic.
is big, this is not viable any more and algorithms with subquadratic runtime are needed.
To avoid checking for edges between every pair of nodes, modern generators use additional data structures that partition the graph into bands.
[5][6] A visualization of this shows a hyperbolic graph with the boundary of the bands drawn in orange.
Points are stored sorted by their angular coordinate in their respective band.
can be (over-)estimated and used to only perform the edge-check for points that lie in a band that intersects the circle.
Additionally, the sorting within each band can be used to further reduce the number of points to look at by only considering points whose angular coordinate lie in a certain range around the one of
(this range is also computed by over-estimating the hyperbolic circle around
), HGGs form an ensemble of networks for which is possible to express the degree distribution analytically as closed form for the limiting case of large number of nodes.
[2] This is worth mentioning since this is not true for many ensembles of graphs.
HGGs have been suggested as promising model for social networks where the hyperbolicity appears through a competition between similarity and popularity of an individual.
