The Fréchet Mean of Inhomogeneous Random Graphs

Abstract

To characterize the “average” of a set of graphs, one can compute the sample Fréchet mean. We prove the following result: if we use the Hamming distance to compute distances between graphs, then the Fréchet mean of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix: an edge exists between the vertices i and j in the Fréchet mean graph if and only if the corresponding entry of the expected adjacency matrix is greater than 1/2. We prove that the result also holds for the sample Fréchet mean when the expected adjacency matrix is replaced with the sample mean adjacency matrix. This novel theoretical result has some significant practical consequences; for instance, the Fréchet mean of an ensemble of sparse inhomogeneous random graphs is the empty graph.