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

Fast Maximum Likelihood estimation via Equilibrium Expectation for Large Network Data