J. Bermond, A. Chaintreau, G. Ducoffe, Dorian Mazauric
{"title":"How long does it take for all users in a social network to choose their communities?","authors":"J. Bermond, A. Chaintreau, G. Ducoffe, Dorian Mazauric","doi":"10.4230/LIPIcs.FUN.2018.6","DOIUrl":null,"url":null,"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). \nPreviously, 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 $\\frac{(2|V|)^{3/2}}{3}$. \nThe 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 $\\Omega(|V|^{\\Theta(\\log{|V|})})$.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"22 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fun with Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.FUN.2018.6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 $\Omega(|V|^{\Theta(\log{|V|})})$.