In this chapter, we study the dynamics of the so-called naming game as an opinion formation model with a focus on how the presence of a set of rigid minorities can result in the emergence of a dominant opinion in the system. These rigid minorities are “speaker-only”, i.e., they only “speak” and never “listen” thus strongly affecting the course of a social agreement process. We show that for a moderate
(fraction of rigid minorities), the agreement dynamics results in an emergence of a dominant opinion. We extensively study the property of such dominant opinions and observe that the dominance is not the characteristic property of only the “speaker-only” opinions; other opinions under certain circumstances can also become dominant. However, with increasing
, the chances of a “speaker-only” opinion becoming dominant increases. We also find early invented opinions possess higher chances of becoming dominant. We embed this model on various static interaction topologies and real-world time-varying face-to-face interaction data. Importantly, for a reasonably static societal structure the presence of rigid minorities influences the emergence of a dominant opinion to a much larger extent than in case where the societal structure is very dynamic.