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

J. Bermond, A. Chaintreau, G. Ducoffe, Dorian Mazauric
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引用次数: 4

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 $\Omega(|V|^{\Theta(\log{|V|})})$.
一个社交网络的所有用户选择他们的社区需要多长时间?
我们考虑社交网络中的社区形成问题,其中用户要么是朋友,要么是敌人。用户被划分为无冲突的组(即,表示用户之间敌意的冲突图$G^- =(V,E)$中的独立集)。只要存在一组最多k个用户(k是任何固定参数),动态就会继续下去,这些用户可以同时改变他们在分区中的当前组,以这种方式,他们都严格增加了他们的效用(朋友数量,即他们各自组的基数减去1)。以前,k $\leq 2$和$O(|V|^3) for k=3$最著名的最大收敛时间上界是$O(|V|\alpha(G^-))$,其中$\alpha(G^-)$是$G^-$的独立数。我们在本文中的第一个贡献在于将初始问题重新解释为整数划分向量上的优势排序的研究。通过这种方法,我们得到$k \leq 2$的紧上界$O(|V| \min\{ \alpha(G^-)$和$\sqrt{|V|} \})$,当$G^-$是空图时,得到阶$\frac{(2|V|)^{3/2}}{3}$的确切值。对于任意固定的k \geq 4,其收敛时间被推测为多项式。在本文中,我们反驳了这一点。具体地,我们证明了对于任意k \geq 4,最大收敛时间为$\Omega(|V|^{\Theta(\log{|V|})})$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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