06.07.2022
Non-Existence of Stable Social Groups in Information-Driven Networks
Erschienen in: Theory of Computing Systems | Ausgabe 4/2022
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Abstract
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Our first result is that when all the social users are either friends, enemies or indifferent to each other (i.e., \(\mathcal {W} = \{-\infty ,0,1\}\)), a partition as above always exists if k ≤ 2, but it may not exist if k ≥ 3. This is in sharp contrast with (Kleinberg and Ligett, Games Econ. Behav. 82, 702–716 2013) who proved that k-stable partitions always exist, for any k, if \(\mathcal {W} = \{-\infty ,1\}\).
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We further study the intriguing relationship between the existence of k-stable partitions and the allowed set of edge-weights \(\mathcal {W}\). Specifically, we give sufficient conditions for the existence or the non existence of such partitions based on tools from Graph Theory. Doing so, we obtain for most sets \(\mathcal {W}\) the largest \(k(\mathcal {W})\) such that all graphs with edge-weights in \(\mathcal {W}\) admit a \(k(\mathcal {W})\)-stable partition.
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From the computational point of view, we prove that for any \(\mathcal {W}\) containing \(-\infty \), the problem of deciding whether a k-stable partition exists is NP-complete for any \(k > k(\mathcal {W})\).