  Public opinion is often affected by the presence of committed groups of
individuals dedicated to competing points of view. Using a model of pairwise
social influence, we study how the presence of such groups within social
networks affects the outcome and the speed of evolution of the overall opinion
on the network. Earlier work indicated that a single committed group within a
dense social network can cause the entire network to quickly adopt the group's
opinion (in times scaling logarithmically with the network size), so long as
the committed group constitutes more than about 10% of the population (with the
findings being qualitatively similar for sparse networks as well). Here we
study the more general case of opinion evolution when two groups committed to
distinct, competing opinions $A$ and $B$, and constituting fractions $p_A$ and
$p_B$ of the total population respectively, are present in the network. We show
for stylized social networks (including Erd\H{o}s-R\'enyi random graphs and
Barab\'asi-Albert scale-free networks) 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. These two regions are separated by two fold-bifurcation
(spinodal) lines which meet tangentially and terminate at a cusp (critical
point). We provide further insights to the phase diagram and to the nature of
the underlying phase transitions by investigating the model on infinite
(mean-field limit), finite complete graphs and finite sparse networks. For the
latter case, we also derive the scaling exponent associated with the
exponential growth of switching times as a function of the distance from the
critical point.
