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2018 | OriginalPaper | Buchkapitel

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

verfasst von : Sonja Smets, Fernando R. Velázquez-Quesada

Erschienen in: Dynamic Logic. New Trends and Applications

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Dynamic Logic: A Personal Perspective

Nächstes Kapitel Dynamic Preference Logic as a Logic of Belief Change

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

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

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.

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.

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.

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 \(b_2\).

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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Titel: The Creation and Change of Social Networks: A Logical Study Based on Group Size
verfasst von: Sonja Smets
Fernando R. Velázquez-Quesada
Verlag: Springer International Publishing
Buch: Dynamic Logic. New Trends and Applications
Print ISBN: 978-3-319-73578-8

Electronic ISBN: 978-3-319-73579-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-73579-5_11

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