Fig. 1. Hyperbolic spaces approximate hierarchical networks.

(A) Example hierarchical description of data and (B) its equivalent representation using Venn diagrams. (C) Venn diagrams can be mapped onto points in a 3D space, forming approximately a tree. The metric in the resulting 3D space is hyperbolic (7). The hyperbolic aspects of the metric are illustrated by the fact that the shortest path between nodes in the tree goes upward and then descends back to the target node. (D) Discrete approximation to a half-space model of the hyperbolic space in 2D. Red solid and dashed lines show the discrete and continuous shortest paths between point a and b within the half-space model of the hyperbolic space (8).