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|α(G−)) for k≤2 and O(|V|3) for k=3, with α(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{α(G−),|V|−−−√}) and, when G− is the empty graph, the exact value of order (2|V|)3/23. The time of convergence, for any fixed k≥4, was conjectured to be polynomial. In this paper we disprove this. Specifically, we prove that for any k≥4, the maximum time of convergence is in Ω(|V|Θ(log|V|)).