  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.
