  The Naming Game is a classic model for studying the emergence and evolution
of language within a population. In this paper, we extend the traditional
Naming Game model to encompass multiple committed opinions and investigate the
system dynamics on the complete graph with an arbitrarily large population and
random networks of finite size. For the fully connected complete graph, the
homogeneous mixing condition enables us to use mean-field theory to analyze the
opinion evolution of the system. However, when the number of opinions
increases, the number of variables describing the system grows exponentially.
To mitigate this, we focus on a special scenario where the largest group of
committed agents competes with a motley of committed groups, each of which is
smaller than the largest one, while initially, most of uncommitted agents hold
one unique opinion. This scenario is chosen for its recurrence in diverse
societies and its potential for complexity reduction by unifying agents from
smaller committed groups into one category. Our investigation reveals that when
the size of the largest committed group reaches the critical threshold, most of
uncommitted agents change their beliefs to this opinion, triggering a phase
transition. Further, we derive the general formula for the multi-opinion
evolution using a recursive approach, enabling investigation into any scenario.
Finally, we employ agent-based simulations to reveal the opinion evolution and
dominance transition in random graphs. Our results provide insights into the
conditions under which the dominant opinion emerges in a population and the
factors that influence these conditions.
