  A wide class of binary-state dynamics on networks---including, for example,
the voter model, the Bass diffusion model, and threshold models---can be
described in terms of transition rates (spin-flip probabilities) that depend on
the number of nearest neighbors in each of the two possible states.
High-accuracy approximations for the emergent dynamics of such models on
uncorrelated, infinite networks are given by recently-developed compartmental
models or approximate master equations (AME). Pair approximations (PA) and
mean-field theories can be systematically derived from the AME. We show that PA
and AME solutions can coincide under certain circumstances, and numerical
simulations confirm that PA is highly accurate in these cases. For monotone
dynamics (where transitions out of one nodal state are impossible, e.g., SI
disease-spread or Bass diffusion), PA and AME give identical results for the
fraction of nodes in the infected (active) state for all time, provided the
rate of infection depends linearly on the number of infected neighbors. In the
more general non-monotone case, we derive a condition---that proves equivalent
to a detailed balance condition on the dynamics---for PA and AME solutions to
coincide in the limit $t \to \infty$. This permits bifurcation analysis,
yielding explicit expressions for the critical (ferromagnetic/paramagnetic
transition) point of such dynamics, closely analogous to the critical
temperature of the Ising spin model. Finally, the AME for threshold models of
propagation is shown to reduce to just two differential equations, and to give
excellent agreement with numerical simulations. As part of this work,
Octave/Matlab code for implementing and solving the differential equation
systems is made available for download.
