  Zealots are the vertices in a social network who do not change their opinions
under social pressure, and are crucial to the study of opinion dynamics on
complex networks. In this paper, we study the effect of zealots on the
polarization dynamics of a deterministic majority-rule model using the
configuration model as a substrate. To this end, we propose a novel quantifier,
called `correlated polarization', for measuring the amount of polarization in
the network when vertices can exists in two opposite states. The quantifier
takes into account not only the fraction of vertices with each opinion, but
also how they are connected to each other. We then show that the presence of
zealots does not have a fixed effect on the polarization, and can change it in
positive, negative or neutral way depending upon their topological
characteristics like degree, their total fraction in the network, density and
degree heterogeneity of the network, and the type of initial conditions of the
dynamics. Our results particularly highlight the importance of the role played
by the initial conditions in drifting the polarization towards lower or higher
values as the total number of zealots is increased.
