The number of realisations of a random graph

TL;DR

This paper investigates the realisation counts of Erdős-Rényi random graphs and complex solutions to a matrix completion problem, revealing they are either infinite or powers of two with computable exponents.

Contribution

It proves that the realisation number of an Erdős-Rényi graph is either infinite or a power of two, with the exponent computable efficiently, and extends similar results to matrix completion solutions.

Findings

Realisation number is either infinity or a power of 2.

Exponent of the power of 2 is computable in polynomial time.

Derived a formula for complex solutions in matrix completion.

Abstract

Determining the number of realisations of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that the $d$-dimensional realisation number of an Erd\H{o}s-Renyi random graph is either infinity or a power of 2 with exponent computable in polynomial time. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ PSD matrix completion problem with randomly-selected non-diagonal unknown entries.