Fig 1.
Estimating Minkowski separations in a graph.
The Minkowski separation between nodes A and F, MAF is approximated as −25 units as 5 is the number of edges in the longest direction-respecting path between them. Nodes B and G are spacelike separated. To estimate this separation we find a pair of points in their mutual past and future. In this case, the only such pair is (A, F). The naive spatial separation between (B, G) is then given by the timelike separation between (A, F) so is +25 units. Note, only the edges not implied by transitivity have been drawn.
Fig 2.
Papers from arXiv embedded in Minkowski space using Lorentzian MDS.
A visualisation of a D = 1 + 1 embedding of the top 2000 most cited papers in the hep-ph citation network, where node size is proportional to the number of citations. Node colour corresponds to publication date, and in both cases this correlates strongly with the time coordinate obtained from the embedding algorithm. The hep-ph citation network appears more broad in space indicating more pairs of papers which are spacelike separated from each other.
Fig 3.
Papers from arXiv embedded in Minkowski space using Lorentzian MDS.
A visualisation of a D = 1 + 1 embedding of the top 2000 most cited papers in the hep-th citation network, where node size is proportional to the number of citations. Node colour corresponds to publication date, and in both cases this correlates strongly with the time coordinate obtained from the embedding algorithm. In contrast to the hep-ph network, the hep-th citation network has most of its papers in a long chain indicating more timelike separated pairs. We highlight the central placement of the most cited paper in that citation network hep-th/9711200, Maldacena’s paper “The Large N Limit of Superconformal Field Theories and Supergravity”. The visually ‘narrow’ citation network of hep-th and ‘broad’ hep-ph agrees with our previous findings in [25].
Fig 4.
Curves showing the sensitivity and specificity of embeddings into D = 2 Minkowski space of causal set graphs in two-, three- and four-dimensional spacetimes, and of a citation network from the hep-th section of arXiv, all with N = 1000 vertices.
Fig 5.
Area under the curve (AUC) values represent the quality of an embedding. Here we show the AUC values for embedding graphs with 2000 nodes into D = 2 Minkowski spacetime. A value of 1 represents a perfect embedding, and a value of 0.5 is random chance. The two-dimensional causal set graph has, as expected, the highest value, since there must be coordinates allowing a perfect embedding (the original coordinates used when building that graph). Higher dimensional causal sets can be embedded less well, but still better than a random DAG (far right). Error bars show the standard deviations of this measurement over 20 randomly generated examples. Notably, the three citation networks we use as examples have significantly higher values that the random DAG illustrating that they have structure which allows a better fit to Minkowski spacetime. The comparatively better fit of the hep-th network over the hep-ph network into 2-dimensional spacetime agrees with our result in [25].