  Full probability models are critical for the statistical modeling of complex
networks, and yet there are few general, flexible and widely applicable
generative methods. We propose a new family of probability models motivated by
the idea of network growth, which we call the Latent Order Logistic (LOLOG)
model. LOLOG is a fully general framework capable of describing any probability
distribution over graph configurations, though not all distributions are easily
expressible or estimable as a LOLOG. We develop inferential procedures based on
Monte Carlo Method of Moments, Generalized Method of Moments and variational
inference. To show the flexibility of the model framework, we show how
so-called scale-free networks can be modeled as LOLOGs via preferential
attachment. The advantages of LOLOG in terms of avoidance of degeneracy, ease
of sampling, and model flexibility are illustrated. Connections with the
popular Exponential-family Random Graph model (ERGM) are also explored, and we
find that they are identical in the case of dyadic independence. Finally, we
apply the model to a social network of collaboration within a corporate law
firm, a friendship network among adolescent students, and the friendship
relations in an online social network.
