Belief Revision Games (BRGs) were recently introduced to simulate the dynamics of beliefs in a network of communicating agents. In a BRG, each agent expresses her beliefs as a propositional formula, which are iteratively revised according to the beliefs of her acquaintances. An appealing property of BRGs is that the belief sequence of each agent is always cyclic and thus can be finitely characterized. However, identifying such belief cycles is a hard task. This paper addresses the computational issues and focuses on the case where the revision policies of the agents are based on a well-known majority-based merging operator. In particular, we show how some evolution patterns in the belief sequences can be identified independently of the propositional language used by the agents to express their beliefs, allowing an exhaustive search of all possible belief cycle patterns. By further identifying beliefs that lead to similar belief cycles, we introduce algorithms to reduce the search space and perform an exhaustive analysis of the dynamics of beliefs in any given network.