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

Authors Jean-Claude Bermond, Augustin Chaintreau, Guillaume Ducoffe, Dorian Mazauric



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Author Details

Jean-Claude Bermond
  • Université Côte d'Azur, CNRS, Inria, I3S, France
Augustin Chaintreau
  • Columbia University in the City of New York, USA
Guillaume Ducoffe
  • National Institute for Research and Development in Informatics and Research Institute of the University of Bucharest, Bucureşti, România
Dorian Mazauric
  • Université Côte d'Azur, Inria, France

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Jean-Claude Bermond, Augustin Chaintreau, Guillaume Ducoffe, and Dorian Mazauric. How long does it take for all users in a social network to choose their communities?. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 6:1-6:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FUN.2018.6

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 <= 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 <= 2 the tight upper-bound O(|V| min {alpha(G^-), sqrt{|V|}}) and, when G^- is the empty graph, the exact value of order ((2|V|)^{3/2})/3. The time of convergence, for any fixed k >= 4, was conjectured to be polynomial [Escoffier et al., 2012][Kleinberg and Ligett, 2013]. In this paper we disprove this. Specifically, we prove that for any k >= 4, the maximum time of convergence is an Omega(|V|^{Theta(log{|V|})}).

Subject Classification

ACM Subject Classification
  • Networks
  • Theory of computation
Keywords
  • communities
  • social networks
  • integer partitions
  • coloring games
  • graphs

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