The number of realisations of a random graph

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 Erdős-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.