Social Network Analysis: Methods and Applications (Structural Analysis in the Social Sciences)

Cover......Page 1
Title page......Page 2
Contents......Page 6
List of Tables......Page 18
List of Illustrations......Page 21
Preface......Page 26
Part I: Networks, Relations, and Structure......Page 30
1 Social Network Analysis in the Social and Behavioral Sciences......Page 32
1.1 The Social Networks Perspective......Page 33
1.2 Historical and Theoretical Foundations......Page 39
1.2.1 Empirical Motivations......Page 40
1.2.2 Theoretical Motivations......Page 42
1.2.3 Mathematical Motivations......Page 44
1.2.4 In Summary......Page 45
1.3 Fundamental Concepts in Network Analysis......Page 46
1.4 Distinctive Features......Page 50
1.5 Organization of the Book aod How to Read It......Page 51
1.5.2 Descriptive and Statistical Methods......Page 52
1.5.4 Chronology......Page 53
1.5.5 Levels of Analysis......Page 54
1.5.6 Chapter Prerequisites......Page 55
1.6 Summary......Page 56
2.1 Introduction: What Are Network Data?......Page 57
2.1.2 Modes......Page 58
2.2 Boundary Specification and Sampling......Page 59
2.2.1 What Is Your Population?......Page 60
2.2.2 Sampling......Page 62
2.3 Types of Networks......Page 64
2.3.1 One-Mode Networks......Page 65
2.3.2 Two-Mode Networks......Page 68
2.3.3 Ego-centered and Special Dyadic Networks......Page 70
2.4.1 Measurement......Page 72
2.4.2 Collection......Page 74
2.4.3 Longitudinal Data Collection......Page 84
2.4.4 Measurement Validity, Reliability, Accuracy, Error......Page 85
2.5 Data Sets Found in These Pages......Page 88
2.5.1 Krackhardt\'s High-tech Managers......Page 89
2.5.2 Padgett\'s Florentine Families......Page 90
2.5.3 Freeman\'s EIES Network......Page 91
2.5.4 Countries Trade Data......Page 93
2.5.5 Galaskiewicz\'s CEOs and Clubs Network......Page 94
2.5.6 Other Data......Page 95
Part II: Mathematical Representations of Social Networks......Page 96
3 Notation for Social Network Data......Page 98
3.1.1 A Single Relation......Page 100
3.1.2 Multiple Relations......Page 102
3.1.3 Summary......Page 104
3.2 Sociometric Notation......Page 106
3.2.1 Single Relation......Page 108
3.2.2 Multiple Relations......Page 110
3.2.3 Summary......Page 112
3.3 Algebraic Notation......Page 113
3.4 Two Sets of Actors......Page 114
3.4.1 Dilferent Types of Pairs......Page 115
3.4.2 Sociometric Notation......Page 116
3.5 Putting It All Together......Page 118
4 Graphs and Matrices......Page 121
4.1 Why Graphs?......Page 122
4.2 Graphs......Page 123
4.2.1 Subgraphs, Dyads, and Triads......Page 126
4.2.2 Nodal Degree......Page 129
4.2.3 Density of Graphs and Subgraphs......Page 130
4.2.4 Example: Padgett\'s Florentine Families......Page 132
4.2.5 Walks, Trails, and Paths......Page 134
4.2.6 Connected Graphs and Components......Page 138
4.2.7 Geodesics, Distance, and Diameter......Page 139
4.2.8 Connectivity of Graphs......Page 141
4.2.9 Isomorphic Graphs and Subgraphs......Page 146
4.2.10 Special Kinds of Graphs......Page 148
4.3 Directed Graphs......Page 150
4.3.1 Subgraphs - Dyads......Page 153
4.3.2 Nodal Indegree and Outdegree......Page 154
4.3.5 Directed Walks, Paths, Semipaths......Page 158
4.3.6 Reachability and Connectivity in Digraphs......Page 161
4.3.8 Special Kinds of Directed Graphs......Page 163
4.4 Signed Graphs and Signed Directed Graphs......Page 165
4.4.1 Signed Graph......Page 166
4.4.2 Signed Directed Graphs......Page 167
4.5 Valued Graphs and Valued Directed Graphs......Page 169
4.5.1 Nodes and Dyads......Page 171
4.5.3 Paths in Valued Graphs......Page 172
4.6 Multigraphs......Page 174
4.7 Hypergraphs......Page 175
4.8.1 Definition......Page 177
4.8.2 Properties of Relations......Page 178
4.9.1 Matrices for Graphs......Page 179
4.9.2 Matrices for Digraphs......Page 181
4.9.3 Matrices for Valued Graphs......Page 182
4.9.6 Basic Matrix Operations......Page 183
4.9.7 Computing Simple Network Properties......Page 188
4.10.1 Reflexivity......Page 193
4.11 Summary......Page 194
Part III: Structural and Locational Properties......Page 196
5 Centrality and Prestige......Page 198
5.1 Prominence: Centrality and Prestige......Page 201
5.1.1 Actor Centrality......Page 202
5.1.2 Actor Prestige......Page 203
5.1.3 Group Centralization and Group Prestige......Page 204
5.2 Nondirectional Relations......Page 206
5.2.1 Degree Centrality......Page 207
5.2.2 Closeness Centrality......Page 212
5.2.3 Betweenness Centrality......Page 217
5.2.4 Information Centrality......Page 221
5.3 Directional Relations......Page 227
5.3.1 Centrality......Page 228
5.3.2 Prestige......Page 231
5.3.3 A Different Example......Page 239
5.4 Comparisons and Extensions......Page 244
6 Structural Balance and Transitivity......Page 249
6.1 Structural Balance......Page 251
6.1.1 Signed Nondirectional Relations......Page 252
6.1.2 Signed Directional Relations......Page 257
6.1.3 Checking for Balance......Page 259
6.1.5 Summary......Page 261
6.2 Clusterability......Page 262
6.2.1 The Clustering Theorems......Page 264
6.2.2 Summary......Page 267
6.3.1 Empirical Evidence......Page 268
6.3.2 Ranked Clusterability......Page 269
6.3.3 Summary......Page 271
6.4 Transitivity......Page 272
6.5 Conclusion......Page 276
7 Cohesive Subgronps......Page 278
7.1.1 Social Group aod Subgroup......Page 279
7.1.2 Notation......Page 281
7.2 Subgroups Based on Complete Mutuality......Page 282
7.2.2 An Example......Page 283
7.2.3 Considerations......Page 285
7.3 Reachability aod Diameter......Page 286
7.3.1 n-cliques......Page 287
7.3.2 An Example......Page 288
7.3.4 n-clans and n-clubs......Page 289
7.3.5 Summary......Page 291
7.4 Subgroups Based on Nodal Degree......Page 292
7.4.1 k-plexes......Page 294
7.4.2 k-cores......Page 295
7.5 Comparing Within to Outside Subgroup Ties......Page 296
7.5.1 LS Sets......Page 297
7.5.2 Lambda Sets......Page 298
7.6 Measures of Subgroup Cohesion......Page 299
7.7.1 Cliques Based on Reciprocated Ties......Page 302
7.7.2 Connectivity in Directional Relations......Page 303
7.7.3 n-cliques in Directional Relations......Page 304
7.8 Valued Relations......Page 306
7.8.1 Cliques, n-cliques, and k-plexes......Page 307
7.8.2 Other Approaches for Valued Relations......Page 311
7.9 Interpretation of Cohesive Subgroups......Page 312
7.10.1 Matrix Permutation Approaches......Page 313
7.10.2 Multidimensional Scaling......Page 316
7.11 Summary......Page 319
8.1 Affiliation Networks......Page 320
8.2.1 Theory......Page 321
8.2.2 Concepts......Page 323
8.2.3 Applications and Rationale......Page 324
8.3.1 The Affiliation Network Matrix......Page 327
8.3.2 Bipartite Graph......Page 328
8.3.3 Hypergraph......Page 332
8.3.5 Summary......Page 335
8.4.1 Definition......Page 336
8.4.2 Examples......Page 338
8.5.1 Properties of Actors and Events......Page 341
8.5.2 Properties of One-mode Networks......Page 343
8.5.3 Taking Account of Subgroup Size......Page 351
8.5.4 Interpretation......Page 353
8.6.1 Galois Lattices......Page 355
8.6.2 Correspondence Analysis......Page 363
! 8.7 Summary......Page 371
Part IV: Roles and Positions......Page 374
9 Structural Eqnivalence......Page 376
9.1.1 Social Roles and Positions......Page 377
9.1.2 An Overview of Positional and Role Analysis......Page 380
9.1.3 A Brief History......Page 383
9.2.1 Definition......Page 385
9.2.2 An Example......Page 386
9.2.3 Some Issues in Defining Structural Equivalence......Page 388
9.3.1 Simplification of Multirelational Networks......Page 390
9.3.2 Tasks in a Positional Analysis......Page 392
9.4 Measuring Structural Equivalence......Page 395
9.4.1 Euclidean Distance as a Measure of Structural Equivalence......Page 396
9.4.2 Correlation as a Measure of Structural Equivalence......Page 397
9.4.3 Some Considerations in Measuring Structural Equivalence......Page 399
9.5.1 Partitioning Actors......Page 404
9.5.2 Spatial Representations of Actor Equivalences......Page 414
9.5.3 Ties Between and Within Positions......Page 417
9.6 Summary......Page 420
10 Blockmodels......Page 423
10.1 Definition......Page 424
10.2 Building Blocks......Page 426
10.2.1 Perfect Fit (Fat Fit)......Page 427
10.2.2 Zeroblock (Lean Fit) Criterion......Page 428
10.2.4 alpha Density Criterion......Page 429
10.2.6 Examples......Page 430
10.2.7 Valued Relations......Page 435
10.3.1 Actor Attributes......Page 437
10.3.2 Describing Individual Positions......Page 440
10.3.3 Image Matrices......Page 446
10.4 Summary......Page 452
11 Relational Algebras......Page 454
11.1 Background......Page 455
11.2 Notation and Algebraic Operations......Page 457
11.2.1 Composition and Compound Relations......Page 458
11.2.2 Properties of Composition and Compound Relations......Page 461
11.3 Multiplication Tables for Relations......Page 462
11.3.1 Multiplication Tables and Relational Structures......Page 464
11.3.2 An Example......Page 468
11.4 Simplification of Role Tables......Page 471
11.4.1 Simplification by Comparing Images......Page 472
11.4.2 Homomorphic Reduction......Page 474
11.5 Comparing Role Structures......Page 478
11.5.1 Joint Homomorphic Reduction......Page 480
11.5.2 The Common Structure Semigroup......Page 481
11.5.3 An Example......Page 482
11.5.4 Measuring the Similarity of Role Structures......Page 486
11.6 Summary......Page 489
12 Network Positions and Roles......Page 490
12.1.1 Theoretical Definitions of Roles and Positions......Page 491
12.1.2 Levels of Role Analysis in Social Networks......Page 493
12.1.3 Equivalences in Networks......Page 495
12.2 Structural Equivalence, Revisited......Page 497
12.3 Automorphic and Isomorphic Equivalence......Page 498
12.3.1 Definition......Page 499
12.3.2 Example......Page 500
12.3.3 Measuring Automorphic Equivalence......Page 501
12.4 Regular Equivalence......Page 502
12.4.1 Definition of Regular Equivalence......Page 503
12.4.2 Regular Equivalence for Nondirectional Relations......Page 504
12.4.3 Regular Equivalence Blockmodels......Page 505
12.4.4 A Measure of Regular Equivalence......Page 508
12.4.5 An Example......Page 510
12.5 \"Types\" of Ties......Page 512
12.5.1 An Example......Page 514
12.6 Local Role Equivalence......Page 516
12.6.1 Measuring Local Role Dissimilarity......Page 517
12.6.2 Examples......Page 520
12.7 Ego Algebras......Page 523
12.7.1 Definition of Ego Algebras......Page 525
12.7.3 Measuring Ego Algebra Similarity......Page 526
12.7.4 Examples......Page 528
12.8 Discussion......Page 531
Part V: Dyadic and Triadic Methods......Page 532
13 Dyads......Page 534
13.1 An Overview......Page 535
13.2 An Example aod Some Definitions......Page 537
13.3 Dyads......Page 539
13.3.1 The Dyad Census......Page 541
13.3.2 The Example and Its Dyad Census......Page 542
13.3.3 An Index for Mutuality......Page 543
13.3.4 A Second Index for Mutuality......Page 547
13.3.5 Subgraph Analysis, in General......Page 549
13.4 Simple Distributions......Page 551
13.4.1 The Uniform Distribution - A Review......Page 553
13.4.2 Simple Distributions on Digraphs......Page 555
13.5 Statistical Analysis of the Number of Arcs......Page 557
13.5.1 Testing......Page 558
13.5.2 Estimation......Page 562
13.6 Conditional Uniform Distributions......Page 564
13.6.1 Uniform Distribution, Conditional on the Number of Arcs......Page 565
13.6.2 Uniform Distribution, Conditional on the Outdegrees......Page 566
13.7 Statistical Analysis of the Number of Mutuals......Page 568
13.7.1 Estimation......Page 569
13.7.2 Testing......Page 571
13.7.3 Examples......Page 572
13.8 Other Conditional Uniform Distributions......Page 573
13.8.1 Uniform Distribution, Conditional on the Indegrees......Page 574
13.8.2 The UlMAN Distribution......Page 576
13.8.3 More Complex Distributions......Page 579
13.9 Other Research......Page 581
13.10 Conclusion......Page 584
14 Triads......Page 585
14.1 Random Models and Substantive Hypotheses......Page 587
14.2 Triads......Page 588
14.2.1 The Triad Census......Page 593
14.2.2 The Example and Its Triad Census......Page 603
14.3 Distribution of a Triad Census......Page 604
14.3.1 Mean and Variance of a k-subgraph Census......Page 605
14.3.2 Mean and Variance of a Triad Census......Page 608
14.3.3 Return to the Example......Page 610
14.3.4 Mean and Variance of Linear Combinations of a Triad Census......Page 611
14.3.5 A Brief Review......Page 613
14.4.1 Configurations......Page 614
14.4.2 From Configurations to Weighting Vectors......Page 619
14.4.3 From Weighting Vectors to Test Statistics......Page 621
14.4.4 An Example......Page 624
14.4.5 Another Example - Testing for Transitivity......Page 625
14.5 Generalizations and Conclusions......Page 627
14.6 Summary......Page 630
Part VI: Statistical Dyadic Interaction Models......Page 632
15 Statistical Analysis of Single Relational Networks......Page 634
15.1 Single Directional Relations......Page 636
15.1.1 The Y-array......Page 637
15.1.2 Modeling the Y -array......Page 641
15.1.3 Parameters......Page 648
15.1.4 Is PI a Random Directed Graph Distribution?......Page 662
15.1.5 Summary......Page 663
15.2 Attribute Variables......Page 664
15.2.1 Introduction......Page 665
15.2.2 The W-array......Page 666
15.2.3 The Basic Model with Attribute Variables......Page 669
15.2.4 Examples: Using Attribute Variables......Page 675
15.3 Related Models for Further Aggregated Data......Page 678
15.3.1 Strict Relational Analysis - The V-array......Page 680
15.3.2 Ordinal Relational Data......Page 683
15.4.1 A Model......Page 685
15.4.2 An Example......Page 686
15.5 Recent Generalizations of PI......Page 687
15.6.1 Introduction......Page 691
15.6.2 The Basic Model......Page 692
15.6.3 Aggregating Dyads for Two-mode Networks......Page 693
15.7 Computing for Log-linear Models......Page 694
15.7.1 Computing Packages......Page 695
15.7.2 From Printouts to Parameters......Page 700
15.8 Summary......Page 702
16 Stochastic Blockmodels and Goodness-of-Fit Indices......Page 704
16.1 Evaluating Blockmodels......Page 707
16.1.1 Goodness-of- Fit Statistics for Blockmodels......Page 708
16.1.2 Structurally Based Blockmodels and Permutation Tests......Page 717
16.1.3 An Example......Page 718
16.2 Stochastic Blockmodels......Page 721
16.2.1 Definition of a Stochastic Blockmodel......Page 723
16.2.2 Definition of Stochastic Equivalence......Page 725
16.2.3 Application to Special Probability Functions......Page 726
16.2.4 Goodness-of-Fit Indices for Stochastic Blockmodels......Page 732
16.2.5 Stochastic a posteriori Blockmodels......Page 735
16.2.6 Measures of Stochastic Equivalence......Page 737
16.2.7 Stochastic Blockmodel Representations......Page 738
16.2.8 The Example Continued......Page 741
16.3.1 Statistical Analysis of Multiple Relational Networks......Page 748
16.3.2 Statistical Analysis of Longitudinal Relations......Page 750
Part VII: Epilogue......Page 754
17.1 Statistical Models......Page 756
17.2 Generalizing to New Kinds of Data......Page 758
17.2.2 Dynamic and Longitudinal Network Models......Page 759
17.3 Data Collection......Page 760
17.5 General Propositions about Structure......Page 761
17.7 Networks and Standard Social and Behavioral Science......Page 762
Appendix A Computer Programs......Page 764
Appendix B Data......Page 767
References......Page 785
Name Index......Page 831
Subject Index......Page 840
List of Notation......Page 848