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DOI: 10.4230/LIPIcs.FUN.2018.6
How long does it take for all users in a social network to choose their communities?

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

Published in: LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

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|})}).
Cite as

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)

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@InProceedings{bermond_et_al:LIPIcs.FUN.2018.6,
  author =	{Bermond, Jean-Claude and Chaintreau, Augustin and Ducoffe, Guillaume and Mazauric, Dorian},
  title =	{{How long does it take for all users in a social network to choose their communities?}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.6},
  URN =		{urn:nbn:de:0030-drops-87972},
  doi =		{10.4230/LIPIcs.FUN.2018.6},
  annote =	{Keywords: communities, social networks, integer partitions, coloring games, graphs}
}
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