Generalized Blockmodeling Patrick Doreian, Vladimir Batagelj and Anuska Ferligoj Frontmatter More Information

Cambridge University Press 0521840856 - Generalized Blockmodeling Patrick Doreian, Vladimir Batagelj and Anuska Ferligoj Frontmatter More information

Generalized Blockmodeling

This book provides an integrated treatment of blockmodeling, the most frequently used technique in social network analysis. It secures its mathematical foundations and then generalizes blockmodeling for analyzing many types of network structures. Examples are used throughout the text and include small group structures, little league baseball teams, intra-organizational networks, inter-organizational networks, baboon grooming networks, marriage ties of noble families, trust networks, signed networks, Supreme Court decisions, journal citation networks, and alliance networks. Also provided is an integrated treatment of algebraic and graph theoretic concepts for network analysis and a broad introduction to cluster analysis. These formal ideas are the foundations for the authors’ proposal for direct optimizational approaches to blockmodeling, which yield blockmodels that best ﬁt the data – a measure of ﬁt that is integral to the establishment of blockmodels – and cre- ates the potential for many generalizations and a deductive use of blockmodeling.

Patrick Doreian is a professor of sociology and statistics at the University of Pittsburgh and is chair of the Department of Sociology. He has edited the Journal of Mathematical Sociology since 1982 and has been a member of the editorial board for Social Networks since 2003. He was a Centennial professor at The London School of Economics during 2002. He has been a visiting professor at the University of California-Irvine and the University of Ljubljana. His interests include social networks, mathematical sociology, interorganizational networks environmental sociology, and social movements. Vladimir Batagelj is a professor of discrete and computational mathematics at the University of Ljubljana and is chair of the Department of Theoretical Computer Science at IMFM, Ljubljana. He is a member of editorial boards of Informatica and Journal of Social Structure. He was visiting professor at University of Pittsburgh in 1990 to 1991 and at University of Konstanz (Germany) in 2002. His main research interests are in graph theory, algorithms on graphs and networks, combinatorial optimization, data analysis, and applications of information technology in educa- tion. He is coauthor (with Andrej Mrvar) of Pajek – a program for analysis and visualization of large networks. AnuˇskaFerligoj is a professor of statistics at the University of Ljubljana and is dean of the Faculty of Social Sciences. She is editor of the series Metodoloski zvezki since 1987 and is a member of the editorial boards of the Journal of Mathematical Sociology, Journal of Classiﬁcation, Social Networks, and Statistics in Transition. She was a Fulbright scholar in 1990 and visiting professor at the University of Pittsburgh. She was awarded the title of Ambassador of Science of the Republic of Slovenia in 1997. Her interests include multivariate analysis (constrained and multicriteria clustering), social networks (measurement quality and blockmodeling), and survey methodology (reliability and validity of measurement).

Structural Analysis in the Social Sciences

The series Structural Analysis in the Social Sciences presents approaches that explain social behavior and institutions by reference to relations among such concrete entities as persons and organizations. This contrasts with at least four other popular strategies used in social science analysis: (1) reductionist attempts to explain a focus solely in terms of individuals; (2) explanations stressing the causal primacy of such abstract concepts as ideas, values, mental harmonies, and cognitive maps (thus, “structuralism” on the Continent should be distinguished from structural analysis in the present sense); (3) technological and material determination; (4) the use of “variables” as the main analytical concept (as in the “structural equation” models that dominated much of the sociology of the 1970s), where the structure connects variables rather than actual social entities. The social network approach is an important example of the strategy of structural analysis. The series also draws on social science theory and research that is not framed explicitly in network terms but rather stresses the importance of relations over the atomization of reduction or the determination of ideas, technology, or material conditions. Although the structural perspective has become extremely popular and inﬂuential in all the social sciences, it does not have a coherent identity, and no series yet pulls together such work under a single rubric. By bringing the achievements of structurally oriented scholars to a wider public, the Structural Analysis series hopes to encourage the use of this very fruitful approach. Mark Granovetter

Other Books in the Series:

1. Mark S. Mizruchi & Michael Schwartz, eds., Intercorporate Relations: The Structural Analysis of Business. 2. Barry Wellman & S. D. Berkowitz, eds., Social Structures: ANetwork Approach . 3. Ronald L. Brieger, ed., Social Mobility and Social Structure. 4. David Knoke, Political Networks: The Structural Perspective. 5. John L. Campbell, J. Rogers Hollingsworth, & Leon N. Lindberg, eds., Governance of the American Economy. 6. Kyriakos Kontopoulos, The Logics of Social Structure. 7. Philippa Pattison, Algebraic Models for Social Structure. 8. Stanley Wasserman & Katherine Faust, Social Network Analysis: Methods and Applications. 9. Gary Herrigel, Industrial Constructions: The Sources of German Industrial Power. 10. Philippe Bourgois, In Search of Respect: Selling Crack in El Barrio. 11. Per Hage & Frank Harary, Island Networks: Communication, Kinship, and Classiﬁcation Struc- tures in Oceana. 12. Thomas Schweizer & Douglas R. White, eds., Kinship, Networks and Exchange. 13. Noah E. Friedkin, AStructural Theory of Social Inﬂuence . 14. David Wank, Commodifying Communism: Business, Trust, and Politics in a Chinese City. 15. Rebecca Adams & Graham Allan, Placing Friendship in Context. 16. Robert L. Nelson & William P. Bridges, Legalizing Gender Inequality: Courts, Markets and Unequal Pay for Women in America. 17. Robert Freeland, The Struggle for Control of the Modern Corporation: Organizational Change at General Motors, 1924–1970. 18. Yi-min Lin, Between Politics and Markets: Firms, Competition, and Institutional Change in Post- Mao China. 19. Nan Lin, Social Capital: ATheory of Social Structure and Action . 20. Christopher Ansell, Schism and Solidarity in Social Movements: The Politics of Labor in the French Third Republic.

Continued after Index

Generalized Blockmodeling

Patrick Doreian University of Pittsburgh

Vladimir Batagelj University of Ljubljana

AnuˇskaFerligoj University of Ljubljana

published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom

cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2005

Printed in the United States of America

Typeface Times Roman 10/12 pt. System LATEX2ε [TB]

Acatalog record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data Doreian, Patrick. Generalized blockmodeling / Patrick Doreian, Vladimir Batagelj, AnuˇskaFerligoj. p. cm. – (Structural analysis in the social sciences) Includes bibliographical references and indexes. ISBN 0-521-84085-6 1. Social networks – Mathematical models. 2. Sociometry. I. Title: Blockmodeling. II. Batagelj, Vladimir, 1948– III. Ferligoj, Anuska. IV. Title. V. Series. HM741.D67 2004 302.015195 – dc22 2004040784

ISBN 0 521 84085 6 hardback

CONTENTS

Preface xiii

1 Social Networks and Blockmodels 1 1.1 An Intuitive Statement of Network Ideas 3 1.1.1 Fundamental Types of Social Relations 5 1.1.2 Types of Relational Data Arrays 11 1.2 Blocks as Parts of Networks 11 1.2.1 Blocks 12 1.3 Some Block Types 14 1.4 Specifying Blockmodels 16 1.4.1 Parent–Child Role Systems 16 1.4.2 Organizational Hierarchies 17 1.4.3 Systems of Ranked Clusters 19 1.4.4 Baboon Grooming Networks 20 1.5 Conventional Blockmodeling 24 1.5.1 Equivalence and Blockmodeling 24 1.6 Generalized Blockmodeling 25 1.7 An Outline Map of the Topics Considered 27 2 Network Data Sets 30 2.1 Classic Data Sets 30 2.1.1 Sampson Monastery Data 31 2.1.2 Bank Wiring Room Data 37 2.1.3 Newcomb Fraternity Data 44 2.2 Newer Data Sets 47 2.2.1 Little League Baseball Teams 47 2.2.2 Political Actor Network 50 2.2.3 Student Government Data 52 2.2.4 Kansas Search and Rescue Network 54 2.2.5 A Bales-Type Group Dynamics Network 56

viii Contents 2.2.6 Ragusan Families Marriage Networks 56 2.2.7 Two Baboon Grooming Networks 60 2.3 Data Set Properties 61 2.4 Some Additional Remarks Concerning Data 63 3 Mathematical Prelude 64 3.1 Basic Set Theory 64 3.2 Relations 70 3.2.1 Operations with Binary Relations 74 3.2.2 Comparing Relations 76 3.2.3 Special Operations 80 3.3 Functions 84 3.3.1 Products of Functions 87 3.3.2 Relational Homomorphisms 88 3.4 Basic Algebra 89 3.5 Transitions to Chapters 4 and 9 93 4 Relations and Graphs for Network Analysis 94 4.1 Graphs 94 4.1.1 Examples of Graphs 104 4.1.2 Traveling on a Graph 107 4.1.3 Graph Coloring 111 4.2 Types of Binary Relations 112 4.2.1 Properties of Relations 113 4.2.2 Closures 114 4.2.3 Computing the Transitive Closure ✄ 115 4.2.4 Special Elements 116 4.2.5 Tournaments ✄ 117 4.3 Partitions and Equivalence Relations 117 4.4 Acyclic Relations 122 4.4.1 Levels 123 4.5 Orders 124 4.5.1 Factorization 125 4.5.2 Hasse Diagram 126 4.5.3 Numberings 127 4.6 Networks 127 4.7 Centrality in Networks 128 4.7.1 Algorithmic Aspects 131 4.8 Summary and Transition 131 5 Clustering Approaches 133 5.1 An Introduction to Cluster Analytic Ideas 133 5.2 Usual Clustering Problems 134 5.2.1 An Example 135 5.2.2 The Usual Steps of Solving Clustering Problems 137

Contents ix 5.3 (Dis)similarities 137 5.3.1 (Dis)similarity Measures for Numerical Data 138 5.3.2 (Dis)similarity Measures for Binary Data 142 5.4 Clustering Algorithms 143 5.4.1 The Hierarchical Approach 144 5.4.2 The Leader Algorithm 149 5.4.3 The Relocation Algorithms 150 5.5 Constrained Clustering 150 5.5.1 The Constrained Clustering Problem 151 5.5.2 Solving Constrained Clustering Problems 154 5.5.3 The Structure Enforcement Coefficient 156 5.5.4 An Empirical Example 156 5.6 Multicriteria Clustering 160 5.6.1 A Multicriteria Clustering Problem 160 5.6.2 Solving Discrete Multicriteria Optimization Problems 161 5.6.3 Direct Multicriteria Clustering Algorithms 161 5.6.4 An Example 164 5.7 Transition to Blockmodeling 167 6 An Optimizational Approach to Conventional Blockmodeling 168 6.1 Conventional Blockmodeling 168 6.1.1 Definitions of Equivalences 170 6.1.2 Equivalence and k-Partite Graphs 176 6.1.3 Establishing Conventional Blockmodels 176 6.1.4 The Indirect Blockmodeling Approach 177 6.1.5 Measuring the Equivalence of Pairs of Units 178 6.2 Optimization and Blockmodeling 184 6.2.1 The Direct Blockmodeling Approach 185 6.2.2 A Criterion for Structural Equivalence 186 6.2.3 A Criterion for Regular Equivalence 187 6.2.4 A Clustering Algorithm 188 6.2.5 Two Artificial Examples 188 6.3 Representing Partitions 192 6.4 Some Empirical Examples 196 6.4.1 Two Little League Baseball Teams 196 6.4.2 The Political Actor Example 201 6.5 An Analysis of a Search and Rescue Operation 203 6.6 Generalized Blockmodeling 209 7 Foundations for Generalized Blockmodeling 210 7.1 Generalization of Equivalences 211 7.1.1 Some Properties of the Predicates 213 7.1.2 Examples 215

x Contents 7.2 Generalized Blockmodeling 220 7.2.1 Blockmodels 220 7.2.2 T -Equivalence 222 7.2.3 Optimization 223 7.3 Two Examples of Generalized Blockmodeling 227 7.3.1 An Artificial Network 227 7.3.2 A Student Government Network 228 7.3.3 Exploring Multiple Partitions 231 7.4 Prespecified Blockmodels 233 7.5 Blockmodel Types 235 7.6 Applications of Prespecified Blockmodels 237 7.6.1 Classroom Liking Ties for Boys and Girls 237 7.6.2 Baboon Grooming Networks 238 7.6.3 Multiple Blockmodels and Inconsistencies 243 7.7 Some Benefits of the Optimization Approach 245 7.8 Extending Generalized Blockmodeling 245 8 Blockmodeling Two-Mode Network Data 247 8.1 Two-Mode Network Data 247 8.2 Approaches to Two-Mode Network Data 248 8.3 Blockmodels for Two-Mode Network Data 249 8.4 A Formalization of Blockmodeling Two-Mode Data 250 8.5 Blockmodels with Empirical Data 251 8.5.1 Supreme Court Voting 251 8.5.2 The Southern Women Event Participation Data 257 8.5.3 Journal-to-Journal Citation Networks 265 8.6 Summary 270 9 Semirings and Lattices 271 9.1 Walks, Paths, and Algebras 271 9.2 Distributivity and Absorption 273 9.2.1 Distributivity 274 9.2.2 Absorption 274 9.3 Valued Graphs 274 9.3.1 Assigning Values to Paths 275 9.3.2 Assessing Paths in Terms of Their Values 276 9.4 Semirings 279 9.4.1 Some Social Network Applications of Semirings 282 9.5 Semilattices and Lattices as Relations 285 9.5.1 Bounds 286 9.5.2 Semilattices and Lattices 287 9.6 Algebraic View on Lattices 290 9.6.1 Types of Lattices 291 9.6.2 Representations 293 9.7 Conclusion 294

Contents xi 10 Balance Theory and Blockmodeling Signed Networks 295 10.1 Structural Balance Theory 296 10.2 Signed Networks 297 10.3 Partitioning Signed Networks and Semirings 299 10.3.1 Examples 301 10.4 A Partitioning Algorithm for Signed Networks 302 10.5 Exactly k-Balanced Structures 304 10.5.1 An Empirical Example 306 10.6 Structures That are Not k-Balanced 307 10.6.1 A Constructed Example 307 10.6.2 An Empirical Example 307 10.7 Another Look at the Bank Wiring Room Data 310 10.8 Balance and Imbalance in a Bales Group 312 10.9 Through-Time Balance Processes 317 10.9.1 The Sampson Data 318 10.9.2 The Newcomb Data 320 10.10 Blockmodeling and Signed Networks 324 11 Symmetric–Acyclic Blockmodels 326 11.1 Blocks for Directed Graphs and Acyclic Graphs 326 11.2 Two Constructed Examples 327 11.3 Establishing Symmetric–Acyclic Decompositions of Networks 328 11.3.1 Ideal Structures 328 11.3.2 Relations without a Symmetric–Acyclic Decomposition 331 11.4 Liking Ties for Children in a Classroom 333 11.5 The Student Government Example 337 11.5.1 A Hypothesized Blockmodel 337 11.5.2 A Second Hypothesized Blockmodel 338 11.6 A Return to the Classroom Example 339 11.7 Marriage Network of the Ragusan Noble Families 340 11.7.1 Network Decomposition 341 11.7.2 Blockmodeling Approach 344 11.8 Discussion 346 12 Extending Generalized Blockmodeling 347 12.1 Block Types 347 12.2 Block Types and Criterion Functions 348 12.3 Using Substantive and Empirical Knowledge 349 12.3.1 Prespeciﬁcation 349 12.3.2 Constraints 350 12.3.3 Imposing Penalties 350 12.4 The Magnitudes of Criterion Functions 350 12.5 The Generalized Blockmodeling Framework 352 12.6 Composition of Blocks 354

xii Contents 12.7 Multiple Fitted Blockmodels 355 12.8 Multiple Relations 356 12.9 Other Networks and Network Types 358 12.10 Network Size and Valued Graphs 360

Bibliography 363 Author Index 375 Subject Index 378

PREFACE

Once upon a time, in 1974 to be precise, three youngsters met during the ECPR workshop at the University of Essex in England. We knew nothing of what would unfold beyond liking each other’s company and sharing a commitment to using mathematical ideas to advance social science research. We stayed in contact and next met at the SOECO conference in Ljubljana in 1976. Over the years since that meeting, we got together when we could, but it was not until the Social Science Methodology conference at Dubrovnik in 1988 that the idea formed for a collaboration. At that time, Vlado and Nuˇsasaw themselves as cluster analysts and Pat was a social network analyst. As we talked, it became clear that the two fields could be joined to mutual benefit. A shared passion and dream was born: form a synthesis of cluster analytic ideas and social network ideas. This book is a result of pursuing that passion and dream. We owe many debts and we will try to acknowledge them here. Institutional support of various forms was critical. A Fulbright fellowship allowed Nuˇsato visit Pittsburgh for seven months in 1990–91, and we thank the Fulbright Commission for their support. The Universities of Ljubljana and Pittsburgh have an exchange agreement. Under that agreement, Vlado came to Pittsburgh in 1991, and we worked on our joint project. This period marked the true beginning of our profes- sional collaboration, and we presented some results at the Sunbelt XI conference in Tampa in 1991. The University of Ljubljana supported Pat as a visiting professor for a two-month stay in 1993, and our first statement on generalized blockmodeling was presented during the International Social Network Conference in Munich later that year. A grant from the National Academy of Science/National Research Council Collaboration in Basic Science and Engineering (COBASE) allowed Pat to return to Ljubljana for another two months in 1994. Under the exchange agreement between our universities, Nuˇsa was able to visit Pittsburgh again in 1996. We thank conference organizers, our funders, and especially our universities for the support that made our joint work possible.

xiv Preface Besides the formal support, Nuˇsaand Vlado made several visits to Pittsburgh and Pat visited Ljubljana in 1998, 1999, 2002 (twice), and 2003. The visits in 2002 were facilitated by Pat’s having a Centennial Professorship at the London School of Economics, and we appreciate that piece of institutional support greatly. Critical as the institutional support has been, our greatest debt is to Esther Sales. Married to Pat, she accepted, supported, and most importantly, encouraged his frequent visits to Slovenia knowing full well that “the block” seemed no more than a pretext for him to go to one of his most favored places on the planet. Socially, we became a quartet and the work of “the triad” could never have proceeded without Esther’s love and support. We are most appreciative of Mark Granovetter’s encouragement and recommen- dation that this book be a part of the Cambridge Structural Analysis series that he edits. Katie Faust reviewed the book for Cambridge and made many wonderful recommendations for changes and restructuring of some of the arguments. (Of course, we did not know who reviewed the manuscript at that time, but by living in the same network, we came to know her identity well before learning it formally.) Peter Marsden generously agreed to read through the manuscript and offered us many helpful suggestions. Although we are convinced of our arguments, it is al- ways nice to receive positive feedback from other scholars, and we appreciate the expressive support as much as the instrumental support. We are also grateful for the contributions of Andrej Mrvar through programming of the 1996 version of the balance partitioning and his part in the creation of PAJEK. He has also been a wonderful supportive colleague over the years. We also value the comments of students in our classes at the Universities of Pittsburgh and Ljubljana, as well as their frankness in telling us when we indulged in havers (Old English for “foolish talk”). There are some less obvious things that made the book possible. We con- sumed much dvojna kava, temno pivo, refoˇsk,teran, shiraz, and Newcastle Brown. We thank many brewers, wine makers, and coffee producers for fueling our endeavors – so much so that we do not envy them their profits from our liquid consumption patterns. We also thank publishers, notably Elsevier and Springer- Verlag, for their permission to use some of the materials that we published in their journals. And we do not envy their profits from our labors either. Some authors are quite forthright in declaring why they write their books and why the world will benefit from someone’s paying attention. We are less ambitious. We wrote this book primarily to have fun, enjoy each other’s company, collect our thoughts in one place, and build upon them. Certainly, our objective was not to become rich and famous! The book is a product of having fun, glorious fun. But we think that we did achieve some worthwhile things. We have produced a synthesis between cluster analysis and network analysis. However, it is for others to judge whether or not this truly has merit. We think that we have secured the mathematical foundations for generalized blockmodeling and have opened the way for many further generalizations. Although the book can be viewed as very technical and mathematical, we do emphasize that substance comes first. This is especially clear

Preface xv in the analysis of role systems, structural balance, and ranked systems. We have a particularly soft spot for the materials in the chapter on the symmetric–acyclic decomposition of networks because for us, the underlying concepts started in sociological substance. The mathematics and methods that we created came in response to substantive concerns instead of groping for something on the shelves. Finally, we like to emphasize the use of “deductive” blockmodeling through the prespeciﬁcation of generalized blockmodels. Often, researchers know more than they think they do, and this knowledge can be mobilized fruitfully. Much remains to be done, and we point to some of this in our ﬁnal chapter. For us, the future concerns us most, and we hope that others will, so to speak, put their shoulders to this wheel. We will continue to work in this area, and the tools we create will be located in PAJEK. Above all, we will continue to have fun and to enjoy the love and affection that we share.

Piran, Slovenia