Π a π represent isomorphic graphs the latter

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Π A Π represent isomorphic graphs, the latter featuring permuted rows and columns of the former. If Π re-indexes nodes according to their categorical groupings, then we may define a conformal partition Π A Π = A 00 A 01 A 01 A 11 that respects this ordering, such that exchangeability is preserved within – but not across – submatrices A 00 and A 11 . We may then simply compute sample proportions corresponding to each submatrix { A 00 , A 01 , A 11 } to yield { p 00 , p 01 , p 11 } . Note that by construction, submatrices A 00 and A 11 yield subgraphs that are themselves Erd¨os–R´ enyi, and are said to be induced by the two respective groups of categorical covariates. Nonzero entries of A 01 are said to comprise the edge boundary between these two induced subgraphs; indeed, the matrix obtained by setting all entries of A 00 and A 11 to zero yields in turn a bipartite graph whose vertices can be partitioned according to their binary covariate values. The following example illustrates these concepts using the simulated data of Example 1.1. Example 1.2 (Similarity and Subgraphs). Let the ten-node network of Example 1.1 be subject to an isomorphism that re-orders nodes accord- ing to the two groups defined by their binary covariate values , and define the permutation-similar data matrix A and permuted covariate vector ˜ c as follows : A = 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 = A 00 A 01 A 01 A 11 ; ˜ c = 0 0 0 0 0 1 1 1 1 1 . 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 9 Fig. 1.2. Subgraphs based on the binary covariates of Example 1.1, again represented graphically by node shape. The conformal partition of Example 1.2 implies two induced subgraphs: solid lines inside the ellipse are links represented in submatrix e A 00 , while those outside it appear as links in e A 11 . The remaining links, shown as dashed lines, correspond to values of 1 in submatrix e A 01 and comprise the associated edge boundary. Figure 1.2 illustrates the corresponding subgraphs using the visualization of Figure 1.1 ; assuming a simple stochastic block model in turn leads to the following maximum-likelihood parameter estimates : p 00 = 5 10 ; p 01 = 7 25 ; p 11 = 2 10 .
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  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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