The Sample Fréchet Mean of Sparse Graphs is Sparse
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The availability of large datasets composed of graphs creates an unprecedented need to invent novel tools in statistical learning for "graph-valued random variables". To characterize the "average" of a sample of graphs, one can compute the sample Fréchet mean. Because the sample mean should provide an interpretable summary of the graph sample, one would expect that the structural properties of the sample be transmitted to the Fréchet mean. In this paper, we address the following foundational question: does the sample Fréchet mean inherit the structural properties of the graphs in the sample? Specifically, we prove the following result: the sample Fréchet mean of a set of sparse graphs is sparse. We prove the result for the graph Hamming distance, and the spectral adjacency pseudometric, using very different arguments. In fact, we prove a stronger result: the edge density of the sample Fréchet mean is bounded by the edge density of the graphs in the sample. This result guarantees that sparsity is an hereditary property, which can be transmitted from a graph sample to its sample Fréchet mean, irrespective of the method used to estimate the sample Fréchet mean.
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