COMPUTER SCIENCE EN
Inference_for_Graphs_and_Networks.pdf

# And it follows that the degree n j 1 a ij of each

• 99

This preview shows pages 6–9. Sign up to view the full content.

, and it follows that the degree n j =1 A ij of each network node is a Binomial( n 1 , p ) random variable. 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

This preview has intentionally blurred sections. Sign up to view the full version.

Inference for Graphs and Networks 7 Fitting the parameter p is straightforward; the maximum likelihood estimator (MLE) corresponds to the sample proportion of observed links: p := 1 ( n 2 ) i<j A ij = 1 n ( n 1) n i =1 n j =1 A ij . Example 1.1, for instance, yields p = 14 / 45. Given a relational data set of interest, we can test the agreement of data in A with this model by employing an appropriately selected test statistic. If we wish to test this uniformly generic model with respect to the notion of network structure, we may explicitly define an alternate model and appeal to the classical Neyman–Pearson testing framework. In this vein, the Erd¨ os–R´ enyi model can be generalized in a natural way to capture the notion of local rather than global exchangeability: we simply allow Bernoulli parameters to depend on k -ary categorical covariates c ( i ) associated with each node i ∈ { 1 , 2 , . . ., n } , where the k n categories represent groupings of nodes. Formally we define c Z n k ; c ( i ) : { 1 , 2 , . . ., n } → Z k , and a set of ( k +1 2 ) distinct Bernoulli parameters governing link probabilities within and between these categories, arranged into a k × k symmetric matrix and indexed as p c ( i ) c ( j ) for i, j ∈ { 1 , 2 , . . ., n } . In the case of binary categorical covariates, we immediately obtain a formulation of Holland and Leinhardt (1981), the simplest example of a so-called stochastic block model . In this network model, pairwise links between nodes correspond again to Bernoulli trials, but with a parameter chosen from the set { p 00 , p 01 , p 11 } according to binary categorical covariates associated with the nodes in question. Definition 1.2 (Simple Stochastic Block Model). Let c ∈ { 0 , 1 } n be a binary n -vector for some integer n > 1 , and fix parameters p 00 , p 01 , p 11 [0 , 1] . Set p 10 = p 01 ; the model then corresponds to matrices A ∈ { 0 , 1 } n × n defined element-wise as i, j ∈ { 1 , 2 , . . ., n } : i < j, A ij Bernoulli( p c ( i ) c ( j ) ); A ji = A ij , A ii = 0 . If the vector of covariates c is given, then finding the maximum- likelihood parameter estimates { p 00 , p 01 , p 11 } is trivial after a re-ordering of 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
8 B. P. Olding and P. J. Wolfe nodes via permutation similarity: For any n × n permutation matrix Π , the adjacency matrices A and

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.
• Spring '12
• Kushal Kanwar
• Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern