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The Creation and Change of Social Networks: A Logical Study Based on Group Size

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Dynamic Logic. New Trends and Applications (DALI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10669))

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Abstract

This paper is part of an on-going programme in which we provide a logical study of social network formations. In the proposed setting, agent a will consider agent b as part of her network if the number of features (properties) on which they differ is small enough, given the constraints on the size of agent a’s ‘social space’. We import this idea about a limit on one’s social space from the cognitive science literature. In this context we study the creation of new networks and use the tools of Dynamic Epistemic Logic to model the updates of the networks. By providing a set of reduction axioms we are able to provide sound and complete axiomatizations for the logics studied in this paper.

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Notes

  1. 1.

    Think, for example, how we establish conversations with relatively ‘distant’ acquaintances mostly only when our close friends are not around.

  2. 2.

    See [7, Chap. 1] for more details on mathematical distances.

  3. 3.

    In such case, and if no additional criteria is used to distinguish agents in the same layer, all of them should ‘stand together’: the decision of whether they will become part of a’s social network should be of a ‘either all or else none’ nature.

  4. 4.

    Numbers over edges indicate distance. Edges in black are actual pairs in the social network relation, and dotted grey edges are shown only for distance information.

  5. 5.

    More precisely, the formula states that there is at least one set of features \(\mathsf {P}'\), of size t, such that a and b differ in all features in \(\mathsf {P}'\) and coincide in all features in \(\mathsf {P}\setminus \mathsf {P}'\). There can be a most one such set; therefore the formula is true exactly when a and b differ in exactly t features.

  6. 6.

    More precisely, the formula states that there are \(j_1, j_2 \in \{ 0, \ldots , \mathopen {\vert } \mathsf {P} \mathclose {\vert } \}\), with \(j_1 \le j_2\), such that \(j_1\) is the distance from a to \(b_1\), and \(j_2\) is the distance from a to \(b_2\).

  7. 7.

    For an example, take a model with \(V(a) = \{ p,q,r \}\), \(V(b_1) = \{ q,r \}\) and \(V(b_2) =\) \(\{ p \}\). Then, \({ \textsc {dist}}^{\{ p,q,r \}}_{M}(a, b_1) = 1 < 2 = { \textsc {dist}}^{\{ p,q,r \}}_{M}(a, b_2)\), but nevertheless \({ \textsc {dist}}^{\{ p \}}_{M}(a, b_2) = 0 < 1 = { \textsc {dist}}^{\{ p \}}_{M}(a, b_1)\).

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Authors and Affiliations

  1. Institute for Logic, Language and Computation, Universiteit van Amsterdam, Amsterdam, The Netherlands

    Sonja Smets & Fernando R. Velázquez-Quesada

Authors
  1. Sonja Smets
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  2. Fernando R. Velázquez-Quesada
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Correspondence to Fernando R. Velázquez-Quesada .

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Editors and Affiliations

  1. INESC TEC, University of Minho, Braga, Portugal

    Alexandre Madeira

  2. UFRJ, Rio de Janeiro, Brazil

    Mário Benevides

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Smets, S., Velázquez-Quesada, F.R. (2018). The Creation and Change of Social Networks: A Logical Study Based on Group Size. In: Madeira, A., Benevides, M. (eds) Dynamic Logic. New Trends and Applications. DALI 2017. Lecture Notes in Computer Science(), vol 10669. Springer, Cham. https://doi.org/10.1007/978-3-319-73579-5_11

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  • DOI: https://doi.org/10.1007/978-3-319-73579-5_11

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