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On the Number of Edges of the Fréchet Mean and Median Graphs

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Network Science (NetSci-X 2022)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 13197))

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Abstract

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 Frechet mean and median graphs. In this paper, we address the following foundational question: does a mean or median graph inherit the structural properties of the graphs in the sample? An important graph property is the edge density; we establish that edge density is an hereditary property, which can be transmitted from a graph sample to its sample Frechet mean or median graphs, irrespective of the method used to estimate the mean or the median. Because of the prominence of the Frechet mean in graph-valued machine learning, this novel theoretical result has some significant practical consequences.

This work was supported by the National Science Foundation, CCF/CIF 1815971.

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Authors and Affiliations

  1. Applied Mathematics, University of Colorado at Boulder, Boulder, CO, 80309, USA

    Daniel Ferguson & François G. Meyer

Authors
  1. Daniel Ferguson
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  2. François G. Meyer
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Correspondence to François G. Meyer .

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Editors and Affiliations

  1. University of Porto, Porto, Portugal

    Pedro Ribeiro

  2. University of Porto, Porto, Portugal

    Fernando Silva

  3. University of Aveiro, Aveiro, Portugal

    José Fernando Mendes

  4. Iscte – Instituto Universitário de Lisboa, Lisbon, Portugal

    Rosário Laureano

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Ferguson, D., Meyer, F.G. (2022). On the Number of Edges of the Fréchet Mean and Median Graphs. In: Ribeiro, P., Silva, F., Mendes, J.F., Laureano, R. (eds) Network Science. NetSci-X 2022. Lecture Notes in Computer Science(), vol 13197. Springer, Cham. https://doi.org/10.1007/978-3-030-97240-0_3

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  • DOI: https://doi.org/10.1007/978-3-030-97240-0_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-97239-4

  • Online ISBN: 978-3-030-97240-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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