title: Effect of zealotry in high-dimensional opinion dynamics models.
abstract: Most of the work on opinion dynamics models focuses on the case of two or three opinion types. We consider the case of an arbitrary number of opinions in the mean field case of the naming game model in which it is assumed the population is infinite and all individuals are neighbors. A particular challenge of the naming game model is that the number of variables, which corresponds to the number of possible sets of opinions, grows exponentially with the number of possible opinions. We present a method for generating mean field dynamical equations for the general case of k opinions. We calculate the steady states in two important special cases in arbitrarily high dimension: the case in which there exist zealots of only one type, and the case in which there are an equal number of zealots for each opinion. We show that in these special cases a phase transition occurs at critical values p(c) of the parameter p describing the fraction of zealots. In the former case, the critical value determines the threshold value beyond which it is not possible for the opinion with no zealots to be held by more nodes than the opinion with zealots, and this point remains fixed regardless of dimension. In the latter case, the critical point p(c) is the threshold value beyond which a stalemate between all k opinions is guaranteed, and we show that it decays precisely as a lognormal curve in k.
authors.affiliation: University of California, Davis, California 95616, USA.
authors.forename: Alex
authors.initials: A
authors.lastname: Waagen
authors.affiliation: U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA.
authors.forename: Gunjan
authors.initials: G
authors.lastname: Verma
authors.affiliation: U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA.
authors.forename: Kevin
authors.initials: K
authors.lastname: Chan
authors.affiliation: U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA.
authors.forename: Ananthram
authors.initials: A
authors.lastname: Swami
authors.affiliation: University of California, Davis, California 95616, USA.|The Santa Fe Institute, Santa Fe, New Mexico 87501, USA.
authors.forename: Raissa
authors.initials: R
authors.lastname: D'Souza