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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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12 B. P. Olding and P. J. Wolfe under an assumed model, then depicting the data set as a network is unhelpful: the data are best summarized as n independent observations of nodes whose connectivity structure is uninformative. In this section we invoke a formal hypothesis testing framework to explore the notion of detecting network structure in greater detail, and propose new approaches that are natural from a statistical point of view but have thus far failed to appear in the literature. To illustrate these ideas we apply three categories of tests to a single data set – that of Section 1.4.1 below – and in turn highlight a number of important topics for further development. 1.4.1. The Zachary karate data Zachary (1977) recorded friendships between 34 members of a collegiate karate club that subsequently split into two groups of size 16 and 18. These data are shown in Figure 1.4, with inter- and intra-group links given in Table 1.1. The network consists of 78 links, with degree sequence (ordered in accordance with the node numbering of Figure 1.4) given by (16 , 9 , 10 , 6 , 3 , 4 , 4 , 4 , 5 , 2 , 3 , 1 , 2 , 5 , 2 , 2 , 2 , 2 , 2 , 3 , 2 , 2 , 2 , 5 , 3 , 3 , 2 , 4 , 3 , 4 , 3 , 6 , 13 , 17), Fig. 1.4. Visualization of the Zachary karate data of Section 1.4.1. Nodes are num- bered and binary categorical covariate values, reflecting the subsequent group split, are indicated by shape. 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 13 Table 1.1. Zachary (1977) karate data. Counts # Links # No Links Total Intra-subgroup: 0–0 33 87 120 Inter-subgroup: 0–1 10 278 288 Intra-subgroup: 1–1 35 118 153 Total 78 483 561 and corresponding sample proportion of observed links given by ˆ p = 78 / ( 34 2 ) = 78 / 561. Sociologists have interpreted the data of Zachary not only as evidence of network structure in this karate club, but also as providing binary cate- gorical covariate values through an indication of the subsequent split into two groups, as per Figure 1.4. This in turn provides us with an opportunity to test various models of network structure – including those introduced in Section 1.3 – with respect to ground truth. 1.4.2. Tests with known categorial covariates We begin by posing the question of whether or not the most basic Erd¨ os– enyi network model of Definition 1.1 – with each node being equally likely to connect to any other node – serves as a good description of the data, given the categorical variable of observed group membership. The classical evaluation of this hypothesis comes via a contingency table test.
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