A visualization of the corresponding network is shown

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A visualization of the corresponding network is shown in Figure 1.1 ; how- ever, note that as no geometric structure is implied by the data set itself , a pictorial rendering such as this is arbitrary and non-unique. Fig. 1.1. The network data of Example 1.1, with nodes indexed by number and binary categorical covariate values by shape. Note that no Euclidean embedding accompanies the data, making visualization a challenging task for large-scale networks. 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 5 In Example 1.1, categorical covariates c ( i ), i ∈ { 1 , 2 , . . ., n } are given; however, in network data sets of practical interest, these covariates may well be latent. This in turn gives rise to many of the principal questions of network inference – in contrast to the traditional setting of relational data. Therefore, the issues of network modeling which arise tend to be distinct; as such, classical approaches (e.g., contingency table tests) are directly applicable to network data only in very restricted circumstances. 1.2.2. Networks as distinct from relational data The main distinction between modern-day network data and classical rela- tional data lies in the requisite computational complexity for inference. Indeed, the computational requirements of large-scale network data sets are substantial. With n nodes we associate ( n 2 ) = n ( n 1) / 2 symmetric relations; beyond this quadratic scaling, latent covariates give rise to a variety of combinatorial expressions in n . Viewed in this light, methods to determine relationships amongst subsets of nodes can serve as an impor- tant tool to “coarsen” network data. In addition to providing a lower- dimensional summary of the data, such methods can serve to increase the computational efficiency of subsequent inference procedures by enabling data reduction and smoothing. The general approach is thus similar to modern techniques for high-dimensional Euclidean data, and indeed may be viewed as a clustering of nodes into groups. From a statistical viewpoint, this notion of subset relations can be conveniently described by a k -ary categorical covariate, with k specify- ing the (potentially latent) model order. By incorporating such a covariate into the probability model for the data adjacency matrix A , the “structure” of the network can be directly tested if this covariate is observed, or instead inferred if latent. It is easily seen that the cardinality of the resultant model space is exponential in the number of nodes n ; even if the cate- gory sizes themselves are given, we may still face a combinatorial inference problem. Thus, even a straightforwardly posed hypothesis test for a rela-
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