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Formal Concept Analysis of Social Networks pp 171–195Cite as

A Formal Concept Analysis Look at the Analysis of Affiliation Networks

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Part of the book series: Lecture Notes in Social Networks ((LNSN))

Abstract

In this paper we try to analyse the dual-projection approach to weighted 2-mode networks using the tools of\(\mathcal{K}\)-Formal Concept Analysis (\(\mathcal{K}\)-FCA), an extension of FCA for incidences with values in a particular kind of semiring.

For this purpose, we first revisit the isomorphisms between 2-mode networks with formal contexts. In the quest for similar relations when the networks have non-Boolean weights, we relate the dual-projection method to both the Singular Value Decomposition and the Eigenvalue Problem of matrices with values in such algebras, as embodied in Kleinberg’s Hubs and Authorities (HITS) algorithm.

To recover a relation with multi-valued extensions of FCA, we introduce extensions of the HITS algorithm to calculate the influence of nodes in a network whose adjacency matrix takes values over dioids, zerosumfree semirings with a natural order. In this way, we show the original HITS algorithm to be a particular instance of the generic construction, but also the advantages of working in idempotent semifields, instances of dioids.

Subsequently, we also make some connections with extended\(\mathcal{K}\)-FCA, where the particular kind of dioid is an idempotent semifield, and provide theoretical reasoning and evidence that the type of knowledge extracted from a matrix by one procedure and the other are different.

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Notes

  1. 1.

    We will consider all graphs in this paper as directed graphs unless otherwise stated.

  2. 2.

    In [9] this is called “a fundamental pattern for the occurrence of lattices in general”; in [8] it is the “Basic Theorem of Concept Lattices”, and in Chap. 3 of [10] it is “the fundamental theorem of concept lattices”.

  3. 3.

    Birkhoff actually coined “crypto-isomorphism”, but the term seems to have been forced to evolve [12]. We point out that the “surprise” must come from finding concepts of different subfields to be the same. Of course cryptomorphisms boil down to plain isomorphisms as soon as the surprise fades away, so it is a mathematical concept more of an educational or sociological than a formal nature.

  4. 4.

    This procedure will be extended in Sect. 2.3.

  5. 5.

    To lessen the visual clutter, we drop the graph index from the matrix.

  6. 6.

    This term is not standard: for instance, [33] prefer to use “semiring with a multiplicative group structure”, but we prefer semifield to shorten out statements.

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Acknowledgements

The authors have been partially supported by the Spanish Government-MinECo projects TEC2014-53390-P and TEC2014-61729-EXP for this work.

We would like to thank the reviewers of earlier versions of this paper for their help in improving it.

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Authors and Affiliations

  1. Department of Signal Theory and Communications, Universidad Carlos III de Madrid, 28911, Leganés, Spain

    Francisco J. Valverde-Albacete & Carmen Peláez-Moreno

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  1. Francisco J. Valverde-Albacete
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  2. Carmen Peláez-Moreno
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Correspondence to Francisco J. Valverde-Albacete .

Editor information

Editors and Affiliations

  1. Department of Computer Science & Engineering, University of Quebec in Outaouais, Quebec, Canada

    Rokia Missaoui

  2. Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia

    Sergei O. Kuznetsov

  3. Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia

    Sergei Obiedkov

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Valverde-Albacete, F.J., Peláez-Moreno, C. (2017). A Formal Concept Analysis Look at the Analysis of Affiliation Networks. In: Missaoui, R., Kuznetsov, S., Obiedkov, S. (eds) Formal Concept Analysis of Social Networks. Lecture Notes in Social Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-64167-6_7

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  • DOI: https://doi.org/10.1007/978-3-319-64167-6_7

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