  The controllability of complex networks may be applicable for understanding
how to control a complex social network, where members share their opinions and
influence one another. Previous works in this area have focused on
controllability, energy cost, or optimization under the assumption that all
nodes are compliant, passing on information neutrally without any preferences.
However, the assumption on nodal neutrality should be reassessed, given that in
networked social systems, some people may hold fast to their personal beliefs.
By introducing some stubborn agents, or zealots, who hold steadfast to their
beliefs and seek to influence others, the control energy is computed and
compared against those without zealots. It was found that the presence of
zealots alters the energy cost at a quadratic rate with respect to their own
fixed beliefs. However, whether or not the zealots' presence increases or
decreases the energy cost is affected by the interplay between different
parameters such as the zealots' beliefs, number of drivers, final control time
regimes, network effects, network dynamics, and number and configurations of
neutral nodes influenced by the zealots. For example, when a network dynamics
is linear but does not have conformity behavior, it could be possible for a
contrarian zealot to assist in reducing control energy. With conformity
behavior, a contrarian zealot always negatively affects network control by
increasing energy cost. The results of this paper suggest caution when modeling
real networked social systems with the controllability of networked linear
dynamics since the system dynamical behavior is sensitive to parameter change.
