  Temporal exponential-family random graph models (TERGMs) are a flexible class
of network models for the dynamics of tie formation and dissolution. In
practice, separable TERGMs (STERGMs) are the subclass most often used, as these
permit estimation from inexpensive cross-sectional study designs, and benefit
from approximations designed to reduce the computational burden. Improving the
approximations are the focus of this paper. We extend the work of Carnegie et
al., which addressed the problem of constructing a STERGM with two specific
equilibrium properties: a cross-sectional distribution defined by a given
exponential-family random graph model (ERGM), and tie durations defined by
given constant hazards of dissolution. We start with Carnegie et al.'s
observation that the exact result is tractable in the dyad-independent case,
and then show that taking the sparse limit of the exact result leads to a
different approximation than the one they presented. We show that the new
approximation outperforms theirs for sparse, dyad-independent models, and that
for dyad-dependent models the errors tend to increase with the level of
dependence for both approximations. We then extend the theoretical results of
Carnegie et al. to the dyad-dependent case, proving that both the old and new
approximations are asymptotically exact as the STERGM time step size goes to
zero, for arbitrary dyad-dependent terms and some dyad-dependent constraints.
We also show that the continuous-time limit of the discrete-time approximations
has exactly the combination of cross-sectional and durational equilibrium
behavior that we seek.
