Embedding graphs in Lorentzian spacetime.

Area of Science:

• Network analysis
• Geometric modeling
• Theoretical physics

Background:

• Geometric network analysis offers powerful descriptive capabilities.
• Classical Multidimensional Scaling (MDS) is limited to Riemannian manifolds.

Purpose of the Study:

• To generalize MDS for any metric signature.
• To develop a method for embedding directed acyclic graphs (DAGs) into Minkowski spacetime.
• To utilize causal structures for spatial and temporal coordinate assignment.

Main Methods:

• Generalization of the classical MDS algorithm to arbitrary metric signatures.
• Development of a novel algorithm exploiting DAG causal structure.
• Embedding DAGs into Minkowski spacetime, mapping causal connections to timelike separation.

Main Results:

• Successfully generalized MDS for broader manifold applications.
• Demonstrated accurate embeddings for causal sets, random DAGs, and citation networks.
• Citation networks showed significantly more accurate embeddings than random DAGs.

Conclusions:

• The geometric embedding method provides a robust framework for network analysis.
• This approach has potential applications in citation analysis, including paper recommendation and identifying missing citations.