hyperbolic_random_graph(pos, beta, r, /, seed=None)#

Return a hyperbolic random undirected graph (also called hyperbolic geometric graph).

The usual hyperbolic random graph model connects pairs of nodes with probability

\[P[(u,v) \in E] = \frac{1}{1+\exp(\beta(d(u,v) - R)/2)},\]

a sigmoid function that decreases as the hyperbolic distance between nodes \(u\) and \(v\) increases. The hyperbolic distance is given by

\[d(u,v) = \text{arccosh}\left[x_0(u) x_0(v) - \sum_{j=1}^D x_j(u) x_j(v) \right],\]

where \(D\) is the dimension of the hyperbolic space and \(x_d(u)\) is the \(d\) th-dimension coordinate of node \(u\) in the hyperboloid model. The number of nodes and the dimension are inferred from the coordinates pos. The 0-dimension “time” coordinate is inferred from the others.

If beta is None, all pairs of nodes with a distance smaller than r are connected.

This algorithm has a time complexity of \(O(n^2)\) for \(n\) nodes.

  • pos (list[list[float]]) – Hyperboloid coordinates of the nodes [[\(x_1(1)\), …, \(x_D(1)\)], [\(x_1(2)\), …, \(x_D(2)\)], …]. The “time” coordinate \(x_0\) is inferred from the other coordinates.

  • beta (float) – Sigmoid sharpness (nonnegative) of the connection probability.

  • r (float) – Distance at which the connection probability is 0.5 for the probabilistic model. Threshold when beta is None.

  • seed (int) – An optional seed to use for the random number generator.


A PyGraph object

Return type: