12 networks as relational data we begin our analysis

Info icon This preview shows pages 3–5. Sign up to view the full content.

View Full Document Right Arrow Icon
1.2. Networks as Relational Data We begin our analysis by making explicit the connection between networks and relational data. In contrast to data sets that may that arise from pair- wise distances or affinities of points in space or time, many modern network data sets are massive, high-dimensional, and non-Euclidean in their struc- ture. We therefore require a way to describe these data other than through purely pictorial or tabular representations – and the notion of cataloging the pairwise relationships that comprise them, with which we begin our analysis, is natural. 1.2.1. Relational data matrices and covariates Graphs provide a canonical representation of relational data as follows: Given n entities or objects of interest with pairs indexed by ( i, j ), we write i j if the i th and j th entities are related, and i j otherwise. These assignments may be expressed by way of an n × n adjacency matrix A , whose entries { A ij } are nonzero if and only if i j . While both the structure of A and the field over which its entries are defined depend on the application or specific data set, a natural connection to graph theory emerges in which entities are represented by vertices, and relations by edges; we adopt the informal but more suggestive descriptors “node” and “link,” respectively. The degree of the i th node is in turn defined as n j =1 A ij . In addition, the data matrix A is often accompanied by covariates c ( i ) associated with each node, i ∈ { 1 , 2 , . . ., n } . Example 1.1 below illustrates a case in which these covariates take the form of binary categorical vari- ables. We shall refer back to these illustrative data throughout Sections 1.2 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
Image of page 3

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

View Full Document Right Arrow Icon
4 B. P. Olding and P. J. Wolfe and 1.3, and later in Section 1.4 will consider a related real-world example: the social network recorded by Zachary (1977), in which nodes repre- sent members of a collegiate karate club and links represent friendships, with covariates indicating a subsequent split of the club into two disjoint groups. Example 1.1 (Network Data Set). As an example data set , consider the ten-node network defined by data matrix A and covariate vector c as A = 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 ; c = 1 0 0 0 1 0 1 1 1 0 .
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern