Sharp Threshold for the Frechet Mean (or Median) of Inhomogeneous Erdos-Renyi Random Graphs

TL;DR

This paper proves that the Frechet mean or median of inhomogeneous Erdos-Renyi random graphs, computed with Hamming distance, is a thresholded graph based on the expected adjacency matrix, resulting in either an empty or complete graph.

Contribution

It establishes a sharp threshold characterization of the Frechet mean and median graphs for inhomogeneous Erdos-Renyi ensembles, linking them to the expected adjacency matrix.

Findings

Frechet mean/median graphs are thresholded versions of the expected adjacency matrix.

The result holds for both population and sample means.

In sparse ensembles, the Frechet mean is always the empty graph.

Abstract

We address the following foundational question: what is the population, and sample, Frechet mean (or median) graph of an ensemble of inhomogeneous Erdos-Renyi random graphs? We prove that if we use the Hamming distance to compute distances between graphs, then the Frechet mean (or median) graph of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix of the ensemble. We show that the result also holds for the sample mean (or median) when the population expected adjacency matrix is replaced with the sample mean adjacency matrix. Consequently, the Frechet mean (or median) graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold: it is either the empty graph, or the complete graph. This novel theoretical result has some significant practical consequences; for instance, the Frechet mean of an ensemble of sparse inhomogeneous random graphs is always the empty graph.