Embedding Graphs in Lorentzian Spacetime

Abstract

Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Euclidean signature, to manifolds of any metric signature. We then use this general method to develop an algorithm to be used on networks which have causal structure allowing them to be embedded in Lorentzian manifolds. The method is demonstrated by calculating embeddings for both causal sets and citation networks in Minkowski spacetime. We finally suggest a number of applications in citation analysis such as paper recommendation, identifying missing citations and fitting citation models to data using this geometric approach.

15 pages, 6 figure, 2 page appendix

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• Order in Innovation
• Navigability of Random Geometric Graphs in the Universe and Other Spacetimes
• Inference of Boundaries in Causal Sets
• Directed Graph Embeddings in Pseudo-Riemannian Manifolds
• HEAT: Hyperbolic Embedding of Attributed Networks
• Embedding Causal Sets into Minkowski Spacetime
• High Performance Algorithms for Quantum Gravity and Cosmology
• Symmetry-driven embedding of networks in hyperbolic space
• Simpler embeddings of causal sets into Minkowski spacetime