A Survey of Statistical Network Models
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Estimation of exponential random graph models for large social networks via graph limits
Proceedings of the 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining
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A variety of random graph models have been developedin recent years to study a range of problems on networks,driven by the wide availability of data from many social,telecommunication, biochemical and other networks. A keymodel, extensively used in the sociology literature, is theexponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, thatis, the larger than expected number of triangles and othersmall subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure.In this paper we characterize the high and low temperatureregimes of the exponential random graph model. Weestablish that in the high temperature regime the mixingtime of the Glauber dynamics is \Theta(n^2 \log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erd˝os-Rényi random graph.