How long does it take for all users in a social network to choose their communities?
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|})}).
communities
social networks
integer partitions
coloring games
graphs
Networks
Theory of computation
6:1-6:21
Regular Paper
https://hal.inria.fr/hal-01780627
Jean-Claude
Bermond
Jean-Claude Bermond
Université Côte d'Azur, CNRS, Inria, I3S, France
Augustin
Chaintreau
Augustin Chaintreau
Columbia University in the City of New York, USA
Guillaume
Ducoffe
Guillaume Ducoffe
National Institute for Research and Development in Informatics and Research Institute of the University of Bucharest, Bucureşti, România
Part of this work has been done as PhD student in the project Coati at Université Côte d'Azur and during visits at Columbia University in the City of New York. This work was also supported by the Institutional research programme PN 1819 Ädvanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD" (2018), project PN 1819-01-01"Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research", funded by the Ministry of Research and Innovation, Romania.
Dorian
Mazauric
Dorian Mazauric
Université Côte d'Azur, Inria, France
Part of this work has been done during his post-doc at Columbia University in the City of New York.
10.4230/LIPIcs.FUN.2018.6
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Jean-Claude Bermond, Augustin Chaintreau, Guillaume Ducoffe, and Dorian Mazauric
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