Figure 1
From: Navigability of Random Geometric Graphs in the Universe and Other Spacetimes
Random geometric graph in (1 + 1)-dimensional de Sitter spacetime. The graph is realized by Poisson sprinkling 700 nodes onto a (1 + 1)-dimensional de Sitter manifold, with compact spatial foliation by circles, which are hypersurfaces of constant time. The temporal cutoff is τ 0 = 5.94, which is the radius of the disk shown. In the figure, the graph has been mapped from the de Sitter manifold to a disk of this radius by equating the time coordinates of all points in de Sitter spacetime with the radial coordinates in the shown disk. A pair of nodes, shown in yellow, is chosen and their light cones are shown in gray and green. The yellow nodes are connected to all other nodes that happen to lie in their corresponding light cones. In particular, the yellow nodes are connected to each other since they lie within each other’s light cones. The overlap between the past and future light cones of the higher-t and lower-t yellow nodes respectively, shown in orange, is their Alexandroff set. The full set of grey links is obtained by iterating over all node pairs.
