The Sample Fr\'echet Mean (or Median) Graph of Sparse Graphs is Sparse

To characterize the "average" of a sample of graphs, one can compute the sample Frechet mean (or median) graph, which provides an interpretable summary of the graph sample. In this paper, we address the following foundational question: does the mean or median graph inherit the structural properties of the graphs in the sample? An important graph property is the edge density. Because sparse graphs provide prototypical models for real networks, one would like to guarantee that the edge density be preserved when computing the sample mean (or median). In this paper, we prove that the edge density is an hereditary property, which can be transmitted from a graph sample to its sample Frechet mean (or median), irrespective of the method used to estimate the mean or the median. Specifically, we prove the following result: the number of edges of the Frechet mean (or median) graph of a set of graphs is bounded by the maximal number of edges amongst all the graphs in the sample. We prove the result for the graph Hamming distance, and the spectral adjacency pseudometric, using very different arguments.