Though the preferential attachment approach serves to

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a Dirichlet process (Pemantle, 2007). Though the preferential attachment approach serves to describe the observed degree sequences of many networks, it can fall short of correctly modeling their patterns of connectivity (Li et al. , 2005); moreover, het- erogenous degree sequences may not necessarily follow power laws. A nat- ural solution to both problems is to condition on the observed degree sequence as in Section 1.4.4 and consider the connections between nodes to be random. As described earlier, the difficulties associated with simulat- ing fixed-degree simple graphs have historically dissuaded researchers from this direction, and hence fixed-degree models have not yet seen wide use in practice. As an alternative to fixed-degree models, researchers have instead focused on the so-called configuration model (Newman et al. , 2001) as well as models which yield graphs of given expected degree (Chung et al. , 2003). 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 23 The configuration model specifies the degree sequence exactly, as with the case of fixed-degree models, but allows both multiple links between nodes and “self-loops” in order to gain a simplified asymptotic analysis. Models featuring given expected degrees specify only the expected degree of each node – typically set equal to the observed degree – and allow the degree sequence to be random. Direct simulation becomes possible if self-loops and multiple links are allowed, thus enabling approximate inference methods of the type described in Section 1.3.2. However, observed network data sets do not always exhibit either of these phenomena, thus rendering the inferen- tial utility of these models highly dependent on context. In the case of very large data sets, for example, the possible presence or absence of multiple connections or self-loops in the model may be irrelevant to describing the data on a coarse scale. When it becomes necessary to model network data at a fine scale, however, a model which allows for these may be insufficiently realistic. Graph models may equally well be tailored to specific fields. For exam- ple, sociologists and statisticians working in concert have developed a class of well-known approaches collectively known as exponential random graph models (ERGMs) or alternatively as p models. Within this class of models, the probability of nodes being linked to each other depends explicitly on parameters that control well-defined sufficient statistics; practitioners draw on sociological theory to determine which connectivity statistics are criti- cal to include within the model. A key advantage of these models is that they can readily incorporate covariates into their treatment of connectivity properties. For a detailed review, along with a discussion of some of the
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  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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