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# How long does it take for all users in a social network to choose their communities?

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
4 ABS - Algorithms, Biology, Structure
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We consider a community formation problem in social networks, where the users are either friends or enemies. The users are partitioned into conflict-free groups (i.e., independent sets in the conflict graph $G^- =(V,E)$ that represents the enmities between users). The dynamics goes on as long as there exists any set of at most k users, k being any fixed parameter, that can change their current groups in the partition simultaneously, in such a way that they all strictly increase their utilities (number of friends i.e., the cardinality of their respective groups minus one). Previously, the best-known upper-bounds on the maximum time of convergence were $O(|V|\alpha(G^-))$ for $k \leq 2$ and $O(|V|^3)$ for k=3, with $\alpha(G^-)$ being the independence number of $G^-$. Our first contribution in this paper consists in reinterpreting the initial problem as the study of a dominance ordering over the vectors of integer partitions. With this approach, we obtain for $k \leq 2$ the tight upper-bound $O(|V| \min\{ \alpha(G^-), \sqrt{|V|} \})$ and, when $G^-$ is the empty graph, the exact value of order $\frac{(2|V|)^{3/2}}{3}$. The time of convergence, for any fixed $k \geq 4$, was conjectured to be polynomial. In this paper we disprove this. Specifically, we prove that for any $k \geq 4$, the maximum time of convergence is in $\Omega(|V|^{\Theta(\log{|V|})})$.
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https://hal.inria.fr/hal-02264327
Contributor : Dorian Mazauric <>
Submitted on : Tuesday, August 6, 2019 - 5:18:32 PM
Last modification on : Friday, December 18, 2020 - 6:46:05 PM
Long-term archiving on: : Thursday, January 9, 2020 - 3:44:07 AM

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Jean-Claude Bermond, Augustin Chaintreau, Guillaume Ducoffe, Dorian Mazauric. How long does it take for all users in a social network to choose their communities?. Discrete Applied Mathematics, Elsevier, 2019, 270, pp.37-57. ⟨10.1016/j.dam.2019.07.023⟩. ⟨hal-02264327⟩

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