Evolution of opinions on social networks in the presence of competing committed groups

Using a model of pairwise social influence, the {\it binary agreement} model (Xie et. al, Phys. Rev. E 84, 011130 (2011)), we study how the presence of two groups of individuals committed to competing opinions, affect the steady-state opinion of influencable individuals on a social network. We assume that two groups committed to distinct opinions $A$ and $B$, and constituting fractions $p_A$, $p_B$ of the total population respectively, are present in the network. We show using mean-field theory that the phase diagram of this system in parameter space $(p_A,p_B)$ consists of two regions, one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. For finite networks (complete graphs, Erd\H{o}s-R\'enyi networks and Barab\'asi-Albert networks), these two regions are separated by two first order transition lines which terminate and meet tangentially at $p_A = p_B \approx 0.1623$, which constitutes a second-order transition point. Finally, we quantify how the exponentially large switching times between steady states in the co-existence region depend on the distance from the second-order transition point for equal committed fractions.