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

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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 an $\Omega(|V|^{\Theta(\log{|V|})})$.
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Dates and versions

hal-01780627 , version 1 (27-04-2018)

Identifiers

  • HAL Id : hal-01780627 , version 1

Cite

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?. FUN 2018 - 9th International Conference on Fun with Algorithms, 2018, La Maddalena, Italy. ⟨hal-01780627⟩
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