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Hermitian Laplacian Operator for Vector Representation of Directed Graphs: An Application to Word Association Norms

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Advances in Computational Intelligence (MICAI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10633))

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Abstract

In this paper, we propose a spectral method for the analysis of directed graphs. For this purpose, a Hermitian Laplacian operator is proposed, that defines interesting properties for the embedding of a graph into a vector space. We use the notions of the Hermitian Laplacian operator to embed a directed graph structure, build over corpura of Word Association Norms, into a vector space. We show that the Hermitian Laplacian operator has advantages over a traditional Laplacian operator when the original structure of the graph is directed. Moreover, we compare the lexical relations obtained by a WAN graph with the connections the Hermitian Laplacian operator establishes between the words of the corpora.

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Notes

  1. 1.

    http://www.jeuxdemots.org/.

  2. 2.

    http://www.smallworldofwords.com.

  3. 3.

    In this paper, we focused in directed graphs; nevertheless, a directed graph is just a special case of a mixed graph. Therefore, when we refer to mixed graphs, it necessarily implies directed graphs.

  4. 4.

    It is pointed that when the graph is undirected this operators are the traditional Adjacency and Laplacian operators [22,23,24].

  5. 5.

    Two things need to be noted: (1) As LH is hermitian all the eigenvalues in the spectrum are real; (2) \(GL_n(\mathbb {C})\) is the General Linear Group of matrices of \(n\times n\) in \(\mathbb {C}\), this is, not invertible matrices.

  6. 6.

    The entropy was determined through a random walk in the graph [34, 35].

  7. 7.

    Precision at \(k, k= 1,2,...\) is a common method in machine translation literature; for example, [45]. In here, we adapt it to evaluate the semantic quality of the embedding word vectors.

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Acknowledgements

Thanks to the project PAPIIT IA400117 “Simulación de normas de asociación de palabras mediante redes de coocurrencias”.

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Authors and Affiliations

  1. Universidad Nacional Autónoma de México, Mexico City, Mexico

    Víctor Mijangos, Gemma Bel-Engux, Natalia Arias-Trejo & Julia B. Barrón-Martínez

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  1. Víctor Mijangos
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  2. Gemma Bel-Engux
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  3. Natalia Arias-Trejo
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  4. Julia B. Barrón-Martínez
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Correspondence to Víctor Mijangos .

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Editors and Affiliations

  1. Universidad Autónoma del Estado de Hidalgo, Pachuca, Mexico

    Félix Castro

  2. INFOTEC Aguascalientes, Aguascalientes, Mexico

    Sabino Miranda-Jiménez

  3. Tecnológico de Monterrey, Atizapán de Zaragoza, Mexico

    Miguel González-Mendoza

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Mijangos, V., Bel-Engux, G., Arias-Trejo, N., Barrón-Martínez, J.B. (2018). Hermitian Laplacian Operator for Vector Representation of Directed Graphs: An Application to Word Association Norms. In: Castro, F., Miranda-Jiménez, S., González-Mendoza, M. (eds) Advances in Computational Intelligence. MICAI 2017. Lecture Notes in Computer Science(), vol 10633. Springer, Cham. https://doi.org/10.1007/978-3-030-02840-4_4

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  • DOI: https://doi.org/10.1007/978-3-030-02840-4_4

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