  A major line of contemporary research on complex networks is based on the
development of statistical models that specify the local motifs associated with
macro-structural properties observed in actual networks. This statistical
approach becomes increasingly problematic as network size increases. In the
context of current research on efficient estimation of models for large network
data sets, we propose a fast algorithm for maximum likelihood estimation (MLE)
that afords a signifcant increase in the size of networks amenable to direct
empirical analysis. The algorithm we propose in this paper relies on properties
of Markov chains at equilibrium, and for this reason it is called equilibrium
expectation (EE). We demonstrate the performance of the EE algorithm in the
context of exponential random graphmodels (ERGMs) a family of statistical
models commonly used in empirical research based on network data observed at a
single period in time. Thus far, the lack of efcient computational strategies
has limited the empirical scope of ERGMs to relatively small networks with a
few thousand nodes. The approach we propose allows a dramatic increase in the
size of networks that may be analyzed using ERGMs. This is illustrated in an
analysis of several biological networks and one social network with 104,103
nodes
