  We present a full stochastic description of the pair approximation scheme to
study binary-state dynamics on heterogeneous networks. Within this general
approach, we obtain a set of equations for the dynamical correlations,
fluctuations and finite-size effects, as well as for the temporal evolution of
all relevant variables. We test this scheme for a prototypical model of opinion
dynamics known as the noisy voter model that has a finite-size critical point.
Using a closure approach based on a system size expansion around a stochastic
dynamical attractor we obtain very accurate results, as compared with numerical
simulations, for stationary and time dependent quantities whether below, within
or above the critical region. We also show that finite-size effects in complex
networks cannot be captured, as often suggested, by merely replacing the actual
system size $N$ by an effective network dependent size $N_{{\rm eff}}$

Physics and Society