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Kernels on Graphs as Proximity Measures

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Algorithms and Models for the Web Graph (WAW 2017)

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Abstract

Kernels and, broadly speaking, similarity measures on graphs are extensively used in graph-based unsupervised and semi-supervised learning algorithms as well as in the link prediction problem. We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This can potentially be useful for recommending the adoption of one or another similarity measure in a machine learning method. Also, we numerically compare various similarity measures in the context of spectral clustering and observe that normalized heat-type similarity measures with log modification generally perform the best.

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Notes

  1. 1.

    M. Saerens [36] has remarked that a more suitable name could be Neumann diffusion kernel, referring to the Neumann series \(\sum _{k=0}^\infty T^k\) (where T is an operator) named after Carl Gottfried Neumann, while a connection of that to John von Neumann is not obvious (the concept of von Neumann kernel in group theory is essentially different).

  2. 2.

    In fact, L. Katz considered \(\sum _{k=1}^\infty (\alpha W)^k.\)

  3. 3.

    For the properties of M-matrices, we refer to [29].

  4. 4.

    If K is symmetric, then (6) coincides with (2).

  5. 5.

    On various alternative versions of the triangle inequality, we refer to [17].

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Acknowledgements

The work of KA and DR was supported by the joint Bell Labs Inria ADR “Network Science” and by UCA-JEDI Idex Grant “HGRAPHS”, and the work of PC was supported by the Russian Science Foundation (project no.16-11-00063 granted to IRE RAS).

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Authors and Affiliations

  1. Inria Sophia Antipolis, Valbonne, France

    Konstantin Avrachenkov & Dmytro Rubanov

  2. RAS Institute of Control Sciences, Moscow, Russia

    Pavel Chebotarev

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  1. Konstantin Avrachenkov
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  2. Pavel Chebotarev
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  3. Dmytro Rubanov
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Correspondence to Konstantin Avrachenkov .

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Editors and Affiliations

  1. Ryerson University, Toronto, Ontario, Canada

    Anthony Bonato

  2. Mathematics, UC San Diego, La Jolla, California, USA

    Fan Chung Graham

  3. Mathematics, Ryerson University, Toronto, Ontario, Canada

    Paweł Prałat

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Avrachenkov, K., Chebotarev, P., Rubanov, D. (2017). Kernels on Graphs as Proximity Measures. In: Bonato, A., Chung Graham, F., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2017. Lecture Notes in Computer Science(), vol 10519. Springer, Cham. https://doi.org/10.1007/978-3-319-67810-8_3

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  • DOI: https://doi.org/10.1007/978-3-319-67810-8_3

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