GLMLE: graph-limit enabled fast computation for fitting exponential random graph models to large social networks

He, Ran; Zheng, Tian

Large network, as a form of big data, has received increasing amount of attention in data science, especially for large social network, which is reaching the size of hundreds of millions, with daily interactions on the scale of billions. Thus analyzing and modeling these data to understand the connectivities and dynamics of large networks is important in a wide range of scientific fields. Among popular models, exponential random graph models (ERGMs) have been developed to study these complex networks by directly modeling network structures and features. ERGMs, however, are hard to scale to large networks because maximum likelihood estimation of parameters in these models can be very difficult, due to the unknown normalizing constant. Alternative strategies based on Markov chain Monte Carlo (MCMC) draw samples to approximate the likelihood, which is then maximized to obtain the maximum likelihood estimators (MLE). These strategies have poor convergence due to model degeneracy issues and cannot be used on large networks. Chatterjee et al. (Ann Stat 41:2428–2461, 2013) propose a new theoretical framework for estimating the parameters of ERGMs by approximating the normalizing constant using the emerging tools in graph theory—graph limits. In this paper, we construct a complete computational procedure built upon their results with practical innovations which is fast and is able to scale to large networks. More specifically, we evaluate the likelihood via simple function approximation of the corresponding ERGM’s graph limit and iteratively maximize the likelihood to obtain the MLE. We also discuss the methods of conducting likelihood ratio test for ERGMs as well as related issues. Through simulation studies and real data analysis of two large social networks, we show that our new method outperforms the MCMC-based method, especially when the network size is large (more than 100 nodes). One limitation of our approach, inherited from the limitation of the result of Chatterjee et al. (Ann Stat 41:2428–2461, 2013), is that it works only for sequences of graphs with a positive limiting density, i.e., dense graphs.



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Social Network Analysis and Mining

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April 2, 2015


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