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We address the following foundational question: what is the population, andsample, Frechet mean (or median) graph of an ensemble of inhomogeneousErdos-Renyi random graphs? We prove that if we use the Hamming distance tocompute distances between graphs, then the Frechet mean (or median) graph of anensemble of inhomogeneous random graphs is obtained by thresholding theexpected adjacency matrix of the ensemble. We show that the result also holdsfor the sample mean (or median) when the population expected adjacency matrixis replaced with the sample mean adjacency matrix. Consequently, the Frechetmean (or median) graph of inhomogeneous Erdos-Renyi random graphs exhibits asharp threshold: it is either the empty graph, or the complete graph. Thisnovel theoretical result has some significant practical consequences; forinstance, the Frechet mean of an ensemble of sparse inhomogeneous random graphsis always the empty graph.

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