Distribution of tabulated statistics they rejected

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distribution of tabulated statistics, they rejected the hypothesis that the friendships they were observing were formed strictly by chance. Asymptotic tests may alleviate the need for simulation in cases of large network data sets, and are available for certain models and test statistics – the χ 2 -test of Section 1.4.2 being one such example. As another example, Holland and Leinhardt (1981) developed asymptotic tests based on likeli- hood ratio statistics to select between different ERGMs. Sociologists and statisticians together have developed results for other test statistics as well, many of which are reviewed by Wasserman and Faust (1994). A desire upon the rejection of a null model, of course, is the fitting of an alternate. However, as demonstrated in Section 1.3.2, direct fitting by max- imum likelihood can prove computationally costly, even for basic network models. A common solution to maximizing the likelihood under an ERGM, for example, is to employ a Markov chain Monte Carlo strategy (Snijders Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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Inference for Graphs and Networks 25 et al. , 2006). Handcock et al. (2007) also used such methods to maximize the likelihood of a latent space network model; additionally, these authors suggested a faster, though approximate, two-stage maximization routine. Other researchers have employed greedy algorithms to maximize the model likelihood. Newman and Leicht (2007) used expectation maximiza- tion (EM) to fit a network model related to stochastic block modeling. Relaxing the precise requirements of the EM algorithm, both Hofman and Wiggins (2008) and Airoldi et al. (2008) have applied a variational Bayes approach (see, e.g., Jordan et al. (1999)) to find maximum likelihood estimates of parameters under a stochastic block model. Reichardt and Bornholdt (2004) applied simulated annealing to maximize the likelihood of network data under a Potts model, a generalization of the Ising model. Rosvall and Bergstrom (2007, 2008) have also employed simulated annealing in network inference in order to maximize information-theoretic functionals of the data. Following any kind of model fitting procedure, a goodness-of-fit test of some kind is clearly desirable. Yet, researchers have thus far struggled to find a clear solution to this problem. Hunter et al. (2008) have proposed a general method of accumulating a wide set of network statistics, and com- paring them graphically to the distribution of these same statistics under a fitted model. Networks which fit well should in turn exhibit few statistics that deviate far from those simulated from the corresponding model.
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