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Public opinion is often affected by the presence of committed groups ofindividuals dedicated to competing points of view. Using a model of pairwisesocial influence, we study how the presence of such groups within socialnetworks affects the outcome and the speed of evolution of the overall opinionon the network. Earlier work indicated that a single committed group within adense social network can cause the entire network to quickly adopt the group'sopinion (in times scaling logarithmically with the network size), so long asthe committed group constitutes more than about 10% of the population (with thefindings being qualitatively similar for sparse networks as well). Here westudy the more general case of opinion evolution when two groups committed todistinct, competing opinions $A$ and $B$, and constituting fractions $p_A$ and$p_B$ of the total population respectively, are present in the network. We showfor stylized social networks (including Erd\H{o}s-R\'enyi random graphs andBarab\'asi-Albert scale-free networks) that the phase diagram of this system inparameter space $(p_A,p_B)$ consists of two regions, one where two stablesteady-states coexist, and the remaining where only a single stablesteady-state exists. These two regions are separated by two fold-bifurcation(spinodal) lines which meet tangentially and terminate at a cusp (criticalpoint). We provide further insights to the phase diagram and to the nature ofthe underlying phase transitions by investigating the model on infinite(mean-field limit), finite complete graphs and finite sparse networks. For thelatter case, we also derive the scaling exponent associated with theexponential growth of switching times as a function of the distance from thecritical point.

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