  The Naming Game is a model of non-equilibrium dynamics for the self-organized
emergence of a linguistic convention or a communication system in a population
of agents with pairwise local interactions. We present an extensive study of
its dynamics on complex networks, that can be considered as the most natural
topological embedding for agents involved in language games and opinion
dynamics. Except for some community structured networks on which metastable
phases can be observed, agents playing the Naming Game always manage to reach a
global consensus. This convergence is obtained after a time generically scaling
with the population's size $N$ as $t\_{conv} \sim N^{1.4 \pm 0.1}$, i.e. much
faster than for agents embedded on regular lattices. Moreover, the memory
capacity required by the system scales only linearly with its size. Particular
attention is given to heterogenous networks, in which the dynamical activity
pattern of a node depends on its degree. High degree nodes have a fundamental
role, but require larger memory capacity. They govern the dynamics acting as
spreaders of (linguistic) conventions. The effects of other properties, such as
the average degree and the clustering, are also discussed.
