Embedding Graphs in Lorentzian Spacetime

TL;DR

This paper introduces a novel method for embedding directed acyclic graphs into Lorentzian spacetime using a generalized multidimensional scaling approach, enabling new geometric analysis of causal networks.

Contribution

It extends classical MDS to manifolds with arbitrary metric signatures and applies this to embed causal and citation networks in Minkowski spacetime.

Findings

Successfully embedded causal sets and citation networks in Minkowski spacetime.

Demonstrated applications include paper recommendation and missing citation identification.

Provided a new geometric framework for analyzing causal network data.

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.