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The Fréchet Mean of Inhomogeneous Random Graphs

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Book cover Complex Networks & Their Applications X (COMPLEX NETWORKS 2021)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1072))

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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.

F. G. Meyer—Supported by the National Science Foundation (CCF/CIF1815971).

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Authors and Affiliations

  1. Applied Mathematics, University of Colorado Boulder, Boulder, CO, 80305, USA

    François G. Meyer

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  1. François G. Meyer
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Correspondence to François G. Meyer .

Editor information

Editors and Affiliations

  1. Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Madrid, Madrid, Spain

    Rosa Maria Benito

  2. IUT Lumière, University of Lyon, Bron Cedex, France

    Chantal Cherifi

  3. LIB, UFR Sciences et Techniques, University of Burgundy, Dijon, France

    Hocine Cherifi

  4. Grupo Interdisciplinar de Sistemas Complejos, Universidad Carlos III de Madrid, Leganés, Madrid, Spain

    Esteban Moro

  5. Thomas J. Watson College of Engineering and Applied Science, Binghamton University, Binghamton, USA

    Luis M. Rocha

  6. Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona, Tarragona, Spain

    Marta Sales-Pardo

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Meyer, F.G. (2022). The Fréchet Mean of Inhomogeneous Random Graphs. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds) Complex Networks & Their Applications X. COMPLEX NETWORKS 2021. Studies in Computational Intelligence, vol 1072. Springer, Cham. https://doi.org/10.1007/978-3-030-93409-5_18

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  • DOI: https://doi.org/10.1007/978-3-030-93409-5_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-93408-8

  • Online ISBN: 978-3-030-93409-5

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