  We propose a new analytical method to study stochastic, binary-state models
on complex networks. Moving beyond the usual mean-field theories, this
alternative approach is based on the introduction of an annealed approximation
for uncorrelated networks, allowing to deal with the network structure as
parametric heterogeneity. As an illustration, we study the noisy voter model, a
modification of the original voter model including random changes of state. The
proposed method is able to unfold the dependence of the model not only on the
mean degree (the mean-field prediction) but also on more complex averages over
the degree distribution. In particular, we find that the degree heterogeneity
---variance of the underlying degree distribution--- has a strong influence on
the location of the critical point of a noise-induced, finite-size transition
occurring in the model, on the local ordering of the system, and on the
functional form of its temporal correlations. Finally, we show how this latter
point opens the possibility of inferring the degree heterogeneity of the
underlying network by observing only the aggregate behavior of the system as a
whole, an issue of interest for systems where only macroscopic, population
level variables can be measured.
