  The Naming Game is an agent-based model where individuals communicate to name
an initially unnamed object. On a large class of networks continual pairwise
interactions lead the system to an ultimate consensus state, in which agents
converge on a globally shared name. Soon after the introduction of the model,
it was observed in literature that on community-based networks the path to
consensus passes through metastable multi-language states. Subsequently, it was
proposed to use this feature as a mean to discover communities in a given
network. In this paper we show that metastable states correspond to genuine
multi-language phases, emerging in the thermodynamic limit when the fraction of
links connecting communities drops below critical thresholds. In particular, we
study the transition to multi-language states in the stochastic block model and
on networks with community overlap. We also examine the scaling of critical
thresholds under variations of topological properties of the network, such as
the number and relative size of communities and the structure of
intra-/inter-community links. Our results provide a theoretical justification
for the proposed use of the model as a community-detection algorithm.
