  In this paper, we derive new continuous time limits of the Utterance
Selection Model (USM) for language change (Baxter et al., Phys. Rev. E {\bf
73}, 046118, 2006). This is motivated by the fact that the Fokker-Planck
continuous time limit derived in the original version of the USM is only valid
for a small range of parameters. We investigate the consequences of relaxing
these constraints on parameters. Using the normal approximation of the
multinomial approximation, we derive a new continuous time limit of the USM in
the form of a weak-noise stochastic differential equation. We argue that this
weak noise, not captured by the Kramers-Moyal expansion, can not be neglected.
We then propose a coarse-graining procedure, which takes the form of a
stochastic version of the \emph{heterogeneous mean field} approximation. This
approximation groups the behaviour of nodes of same degree, reducing the
complexity of the problem. With the help of this approximation, we study in
detail two simple families of networks: the regular networks and the
star-shaped networks. The analysis reveals and quantifies a finite size effect
of the dynamics. If we increase the size of the network by keeping all the
other parameters constant, we transition from a state where conventions emerge
to a state when no convention emerges. Furthermore, we show that the degree of
a node acts as a time scale. For heterogeneous networks such as star-shaped
networks, the time scale difference can become very large leading to a noisier
behaviour of highly connected nodes.
