For a detailed review along with a discussion of some

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properties. For a detailed review, along with a discussion of some of the latest developments in the field of social networks, the reader is referred to Anderson et al. (1999) and Snijders et al. (2006). Since their original introduction, ERGMs have been widely adopted as models for social networks. They have not yet, however, been embraced to the same extent by researchers outside of social network analysis. Sociolo- gists can rely on existing theory to select models for how humans form rela- tionships with each other; researchers in other fields, though, often cannot appeal to equivalent theories. For exploratory analysis, they may require more generic models to describe their data, appearing to prefer models with a latent vector of covariates to capture probabilistically exchange- able blocks. Indeed, as noted in Section 1.3.1, this approach falls under the general category of stochastic block modeling. Wang and Wong (1987) detail similarities and differences between this approach and the original specification of ERGMs. 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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24 B. P. Olding and P. J. Wolfe Stochastic block modeling, though relatively generic, may still fail to adequately describe networks in which nodes roughly group together, yet in large part fail to separate into distinct clusters. In cases such as this, where stochastic exchangeability is too strong an assumption, standard block modeling breaks down. To this end, two possible modeling solutions have been explored to date in the literature. Hoff et al. (2002) introduced a latent space approach, describing the probability of connection as a function of distances between nodes in an unobserved space of known dimensionality. In this model, the observed grouping of nodes is a result of their proximity in this latent space. In contrast, Airoldi et al. (2007) retained the explicit grouping structure that stochastic block modeling provides, but introduced the idea of mixed group membership to describe nodes that fall between groups. Node membership here is a vector describing partial membership in all groups, rather than an indicator variable specifying a single group membership. A.2. Model Fitting and Inference Even when a model or class of models for network data can be specified, realizing inference can be challenging. One of the oldest uses of random graph models is as a null; predating the computer, Moreno and Jennings (1938) simulated a random graph model quite literally by hand in order to tabulate null model statistics. These authors drew cards out of a ballot shuffling apparatus to generate graphs of the same size as a social network of schoolgirls they had observed. Comparing the observed statistics to the
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