  The analysis of datasets taking the form of simple, undirected graphs
continues to gain in importance across a variety of disciplines. Two choices of
null model, the logistic-linear model and the implicit log-linear model, have
come into common use for analyzing such network data, in part because each
accounts for the heterogeneity of network node degrees typically observed in
practice. Here we show how these both may be viewed as instances of a broader
class of null models, with the property that all members of this class give
rise to essentially the same likelihood-based estimates of link probabilities
in sparse graph regimes. This facilitates likelihood-based computation and
inference, and enables practitioners to choose the most appropriate null model
from this family based on application context. Comparative model fits for a
variety of network datasets demonstrate the practical implications of our
results.
