Generalized Blockmodeling,9780521840859

Generalized Blockmodeling

by
ISBN13: 9780521840859
ISBN10: 0521840856
Format: Hardcover
Pub. Date: 2004-11-08
Publisher(s): Cambridge University Press
List Price: $97.65

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Summary

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 the analysis of 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 creates the potential for many generalizations and a deductive use of blockmodeling.

Table of Contents

Preface xiii
1 Social Networks and Blockmodels
1(29)
1.1 An Intuitive Statement of Network Ideas
3(8)
1.1.1 Fundamental Types of Social Relations
5(6)
1.1.2 Types of Relational Data Arrays
11(1)
1.2 Blocks as Parts of Networks
11(3)
1.2.1 Blocks
12(2)
1.3 Some Block Types
14(2)
1.4 Specifying Blockmodels
16(8)
1.4.1 Parent-Child Role Systems
16(1)
1.4.2 Organizational Hierarchies
17(2)
1.4.3 Systems of Ranked Clusters
19(1)
1.4.4 Baboon Grooming Networks
20(4)
1.5 Conventional Blockmodeling
24(1)
1.5.1 Equivalence and Blockmodeling
24(1)
1.6 Generalized Blockmodeling
25(2)
1.7 An Outline Map of the Topics Considered
27(3)
2 Network Data Sets
30(34)
2.1 Classic Data Sets
30(17)
2.1.1 Sampson Monastery Data
31(6)
2.1.2 Bank Wiring Room Data
37(7)
2.1.3 Newcomb Fraternity Data
44(3)
2.2 Newer Data Sets
47(14)
2.2.1 Little League Baseball Teams
47(3)
2.2.2 Political Actor Network
50(2)
2.2.3 Student Government Data
52(2)
2.2.4 Kansas Search and Rescue Network
54(2)
2.2.5 A Bales-Type Group Dynamics Network
56(1)
2.2.6 Ragusan Families Marriage Networks
56(4)
2.2.7 Two Baboon Grooming Networks
60(1)
2.3 Data Set Properties
61(2)
2.4 Some Additional Remarks Concerning Data
63(1)
3 Mathematical Prelude
64(30)
3.1 Basic Set Theory
64(6)
3.2 Relations
70(14)
3.2.1 Operations with Binary Relations
74(2)
3.2.2 Comparing Relations
76(4)
3.2.3 Special Operations
80(4)
3.3 Functions
84(5)
3.3.1 Products of Functions
87(1)
3.3.2 Relational Homomorphisms
88(1)
3.4 Basic Algebra
89(4)
3.5 Transitions to Chapters 4 and 9
93(1)
4 Relations and Graphs for Network Analysis
94(39)
4.1 Graphs
94(18)
4.1.1 Examples of Graphs
104(3)
4.1.2 Traveling on a Graph
107(4)
4.1.3 Graph Coloring
111(1)
4.2 Types of Binary Relations
112(5)
4.2.1 Properties of Relations
113(1)
4.2.2 Closures
114(1)
4.2.3 Computing the Transitive Closure
115(1)
4.2.4 Special Elements
116(1)
4.2.5 Tournaments
117(1)
4.3 Partitions and Equivalence Relations
117(5)
4.4 Acyclic Relations
122(2)
4.4.1 Levels
123(1)
4.5 Orders
124(3)
4.5.1 Factorization
125(1)
4.5.2 Hasse Diagram
126(1)
4.5.3 Numberings
127(1)
4.6 Networks
127(1)
4.7 Centrality in Networks
128(3)
4.7.1 Algorithmic Aspects
131(1)
4.8 Summary and Transition
131(2)
5 Clustering Approaches
133(35)
5.1 An Introduction to Cluster Analytic Ideas
133(1)
5.2 Usual Clustering Problems
134(3)
5.2.1 An Example
135(2)
5.2.2 The Usual Steps of Solving Clustering Problems
137(1)
5.3 (Dis)similarities
137(6)
5.3.1 (Dis)similarity Measures for Numerical Data
138(4)
5.3.2 (Dis)similarity Measures for Binary Data
142(1)
5.4 Clustering Algorithms
143(7)
5.4.1 The Hierarchical Approach
144(5)
5.4.2 The Leader Algorithm
149(1)
5.4.3 The Relocation Algorithms
150(1)
5.5 Constrained Clustering
150(10)
5.5.1 The Constrained Clustering Problem
151(3)
5.5.2 Solving Constrained Clustering Problems
154(2)
5.5.3 The Structure Enforcement Coefficient
156(1)
5.5.4 An Empirical Example
156(4)
5.6 Multicriteria Clustering
160(7)
5.6.1 A Multicriteria Clustering Problem
160(1)
5.6.2 Solving Discrete Multicriteria Optimization Problems
161(1)
5.6.3 Direct Multicriteria Clustering Algorithms
161(3)
5.6.4 An Example
164(3)
5.7 Transition to Blockmodeling
167(1)
6 An Optimizational Approach to Conventional Blockmodeling
168(42)
6.1 Conventional Blockmodeling
168(16)
6.1.1 Definitions of Equivalences
170(6)
6.1.2 Equivalence and k-Partite Graphs
176(1)
6.1.3 Establishing Conventional Blockmodels
176(1)
6.1.4 The Indirect Blockmodeling Approach
177(1)
6.1.5 Measuring the Equivalence of Pairs of Units
178(6)
6.2 Optimization and Blockmodeling
184(8)
6.2.1 The Direct Blockmodeling Approach
185(1)
6.2.2 A Criterion for Structural Equivalence
186(1)
6.2.3 A Criterion for Regular Equivalence
187(1)
6.2.4 A Clustering Algorithm
188(1)
6.2.5 Two Artificial Examples
188(4)
6.3 Representing Partitions
192(4)
6.4 Some Empirical Examples
196(7)
6.4.1 Two Little League Baseball Teams
196(5)
6.4.2 The Political Actor Example
201(2)
6.5 An Analysis of a Search and Rescue Operation
203(6)
6.6 Generalized Blockmodeling
209(1)
7 Foundations for Generalized Blockmodeling
210(37)
7.1 Generalization of Equivalences
211(9)
7.1.1 Some Properties of the Predicates
213(2)
7.1.2 Examples
215(5)
7.2 Generalized Blockmodeling
220(7)
7.2.1 Blockmodels
220(2)
7.2.2 Τ-Equivalence
222(1)
7.2.3 Optimization
223(4)
7.3 Two Examples of Generalized Blockmodeling
227(6)
7.3.1 An Artificial Network
227(1)
7.3.2 A Student Government Network
228(3)
7.3.3 Exploring Multiple Partitions
231(2)
7.4 Prespecified Blockmodels
233(2)
7.5 Blockmodel Types
235(2)
7.6 Applications of Prespecified Blockmodels
237(8)
7.6.1 Classroom Liking Ties for Boys and Girls
237(1)
7.6.2 Baboon Grooming Networks
238(5)
7.6.3 Multiple Blockmodels and Inconsistencies
243(2)
7.7 Some Benefits of the Optimization Approach
245(1)
7.8 Extending Generalized Blockmodeling
245(2)
8 Blockmodeling Two-Mode Network Data
247(24)
8.1 Two-Mode Network Data
247(1)
8.2 Approaches to Two-Mode Network Data
248(1)
8.3 Blockmodels for Two-Mode Network Data
249(1)
8.4 A Formalization of Blockmodeling Two-Mode Data
250(1)
8.5 Blockmodels with Empirical Data
251(19)
8.5.1 Supreme Court Voting
251(6)
8.5.2 The Southern Women Event Participation Data
257(8)
8.5.3 Journal-to-Journal Citation Networks
265(5)
8.6 Summary
270(1)
9 Semirings and Lattices
271(24)
9.1 Walks, Paths, and Algebras
271(2)
9.2 Distributivity and Absorption
273(1)
9.2.1 Distributivity
274(1)
9.2.2 Absorption
274(1)
9.3 Valued Graphs
274(5)
9.3.1 Assigning Values to Paths
275(1)
9.3.2 Assessing Paths in Terms of Their Values
276(3)
9.4 Semirings
279(6)
9.4.1 Some Social Network Applications of Semirings
282(3)
9.5 Semilattices and Lattices as Relations
285(5)
9.5.1 Bounds
286(1)
9.5.2 Semilattices and Lattices
287(3)
9.6 Algebraic View on Lattices
290(4)
9.6.1 Types of Lattices
291(2)
9.6.2 Representations
293(1)
9.7 Conclusion
294(1)
10 Balance Theory and Blockmodeling Signed Networks 295(31)
10.1 Structural Balance Theory
296(1)
10.2 Signed Networks
297(2)
10.3 Partitioning Signed Networks and Semirings
299(3)
10.3.1 Examples
301(1)
10.4 A Partitioning Algorithm for Signed Networks
302(2)
10.5 Exactly kappa-Balanced Structures
304(3)
10.5.1 An Empirical Example
306(1)
10.6 Structures That are Not k-Balanced
307(3)
10.6.1 A Constructed Example
307(1)
10.6.2 An Empirical Example
307(3)
10.7 Another Look at the Bank Wiring Room Data
310(2)
10.8 Balance and Imbalance in a Bales Group
312(5)
10.9 Through-Time Balance Processes
317(29)
10.9.1 The Sampson Data
318(2)
10.9.2 The Newcomb Data
320(4)
10.10 Blockmodeling and Signed Networks
324(2)
11 Symmetric-Acyclic Blockmodels 326(21)
11.1 Blocks for Directed Graphs and Acyclic Graphs
326(1)
11.2 Two Constructed Examples
327(1)
11.3 Establishing Symmetric-Acyclic Decompositions of Networks
328(5)
11.3.1 Ideal Structures
328(3)
11.3.2 Relations without a Symmetric-Acyclic Decomposition
331(2)
11.4 Liking Ties for Children in a Classroom
333(4)
11.5 The Student Government Example
337(2)
11.5.1 A Hypothesized Blockmodel
337(1)
11.5.2 A Second Hypothesized Blockmodel
338(1)
11.6 A Return to the Classroom Example
339(1)
11.7 Marriage Network of the Ragusan Noble Families
340(6)
11.7.1 Network Decomposition
341(3)
11.7.2 Blockmodeling Approach
344(2)
11.8 Discussion
346(1)
12 Extending Generalized Blockmodeling 347(16)
12.1 Block Types
347(1)
12.2 Block Types and Criterion Functions
348(1)
12.3 Using Substantive and Empirical Knowledge
349(1)
12.3.1 Prespecification
349(1)
12.3.2 Constraints
350(1)
12.3.3 Imposing Penalties
350(1)
12.4 The Magnitudes of Criterion Functions
350(2)
12.5 The Generalized Blockmodeling Framework
352(2)
12.6 Composition of Blocks
354(1)
12.7 Multiple Fitted Blockmodels
355(1)
12.8 Multiple Relations
356(2)
12.9 Other Networks and Network Types
358(2)
12.10 Network Size and Valued Graphs
360(3)
Bibliography 363(12)
Author Index 375(3)
Subject Index 378

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