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Vertex-Frequency Analysis of Graph Signals pp 111–139Cite as

Transformation from Graphs to Signals and Back

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Part of the book series: Signals and Communication Technology ((SCT))

Abstract

Network science has been a rapidly evolving field to study systems made of interactions between entities. Studying the structure of such networks reveals indeed the underlying mechanisms of these systems, and has been proven successful in many domains, such as sociology, biology, or geography. Recently, connections between network science and signal processing have emerged, making the use of a wide variety of tools possible to study networks. In this chapter, a focus is made on a methodology introduced to transform a graph into a collection of signals, using a multidimensional scaling technique: by projecting a distance matrix representing relations between vertices of the graph as points in a Euclidean space, it is possible to interpret coordinates of vertices in this space as signals, and take advantage of this dual representation to develop new tools for the study of networks. Deeper considerations of this methodology are proposed, by strengthening the connections between the obtained signals and the common graph structures. A robust inverse transformation method is next described, taking into account possible changes in the signals. Establishing a robust duality between graphs and signals opens up new perspectives, as classical signal processing tools, such as spectral analysis or filtering, are made available for the study of the structure of networks.

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Acknowledgements

Figures displayed in this article have been created using Python and the following libraries: matplotlib/seaborn [27, 56], networkx [20], numpy/scipy [53], and sklearn [42]. All the code is available at the following url: https://github.com/r-hamon/pygas.

This work was supported by the programs ARC 5 and ARC 6 of the région Rhône-Alpes, and the ANR projects Vél’Innov ANR-12-SOIN-0001-02 and GRAPHSIP ANR-14-CE27-0001.

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Authors and Affiliations

  1. Laboratoire de Physique, University of Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 69342, Lyon, France

    Ronan Hamon, Pierre Borgnat & Patrick Flandrin

  2. University of Lyon, Insa Lyon, Univ Claude Bernard, CNRS, LIRIS, 69342, Lyon, France

    Céline Robardet

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  1. Ronan Hamon
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  2. Pierre Borgnat
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  4. Céline Robardet
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Correspondence to Ronan Hamon .

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Editors and Affiliations

  1. Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

    Ljubiša Stanković

  2. Department of Electrical and Computer Engineering, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Ervin Sejdić

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Hamon, R., Borgnat, P., Flandrin, P., Robardet, C. (2019). Transformation from Graphs to Signals and Back. In: Stanković, L., Sejdić, E. (eds) Vertex-Frequency Analysis of Graph Signals. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-03574-7_2

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  • DOI: https://doi.org/10.1007/978-3-030-03574-7_2

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