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Asymmetrical triadic relationship based on the structural difficulty

A Bayesian approach for balance theory

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Abstract

Using Bayesian inference, this study aims to estimate the magnitude of the cognitive load when a person perceives asymmetric social relations. Some empirical evidence relating to balance theory has shown that a balanced state is comparatively easier to memorize than an unbalanced one. In this study, since a balanced state is defined by structural complexity, an experimental hypothesis was set whereby asymmetric social relationships have different difficulty levels depending on structural complexity. The balanced state of an asymmetric relation as structural difficulty is formally derived from the eigenvalue structure of a Hermitian matrix. Asymmetric triadic relations are modeled as featuring three kinds of structural difficulties according to the eigenvalue decomposition of the Hermitian matrix and pattern-specific difficulties. The differences among the structural difficulties were not sufficiently significant to exceed pattern-specific difficulties, but the Bayes factor of the informational hypothesis of this research yielded positive effects.

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Notes

  1. For individual differences \(\theta \), and a vaguelle uniform distribution such as \({\text {Uniform}}(-100,100)\), this model will not converge. However, it is a natural assumption to use a normal distribution for individual difference, rather than a standard normal distribution. Even if a standard normal distribution is assumed, numerical values such as the Bayes factor described below would change, but research hypothesis shown in this paper remains unchanged and consistent results are obtained.

  2. The complexity of the hypothesis is, for example, in the case of this study, \(H_1\) places two inequality constraints out of the three parameters, and since the combination is \(3! = 6\), one of them complicates \(c_1 = 1/6 \). Similarly, it is 1/3 of the complexity for the hypothesis \(H_2, H_3\) (Okada 2014).

  3. The details are provided by Kruschke (2014).

  4. See the following URL: https://osf.io/fnnk4/

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Acknowledgements

This work was supported by JSPS KAKENHI Grant number 16K13459 and special Thanks to Dr. Hiroshi Shimizu (Kwansei Gakuin Univ.) and Dr. Hirakawa Makoto (Hiroshima Univ.).

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Authors and Affiliations

  1. Faculty of Education, Yamaguchi University, Yoshida 1677-1, Yamaguchi, Yamaguchi, Japan

    Koji E. Kosugi

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  1. Koji E. Kosugi
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Correspondence to Koji E. Kosugi.

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Communicated by: Kensuke Okada.

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Kosugi, K.E. Asymmetrical triadic relationship based on the structural difficulty. Behaviormetrika 45, 7–23 (2018). https://doi.org/10.1007/s41237-017-0033-9

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  • DOI: https://doi.org/10.1007/s41237-017-0033-9

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