Although several models in the literature analyze the dynamics of opinion formation, less attention has been paid to explain how the structure of social networks and their contextual circumstances can influence the course of minority public opinions. This work aims to pose three basic questions: (1) how the structure of social networks can affect the spread of minority opinion, (2) how committed agents influence this process, and (3) how mass media action, as a contextual factor, can vary different agents’ opinions and network composition. Agent-based modeling is used to create a network model of preferential attachment to explore how phenomena of minority opinion spreading can evolve under different simulated scenarios. This study shows that the success of minority opinions depends on network structure and composition and thus on external factors such as mass media action that can mediate the strength of these internal determinants. Although people tend to remain silent when they feel that their opinions are in the minority, our findings suggest that prevailing majority opinion may be promptly replaced by what was formerly minority opinion if core agents in the network structure and/or external sources support this view.