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E-raamat: Algebraic Analysis of Social Networks: Models, Methods and Applications Using R

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Algebraic Analysis of Social Networks: Models, Methods and Applications Using RSuurem pilt
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"Presented in a comprehensive manner, this book provides a comprehensive foundation in algebraic approaches for the analysis of different types of social networks such as multiple, signed, and affiliation networks. The study of such configurations corresponds to the structural analysis within the social sciences, and the methods applied for the analysis are in the areas of abstract algebra, combinatorics, and graph theory. Current research in social networks has moved toward the examination of more realistic but also more complex social relations by which agents or actors are connected in multiple ways. Addressing this trend, this book offers hands-on training of the algebraic procedures presented along with the computer package multiplex, written by thebook's author specifically to perform analyses of multiple social networks. An introductory section on both complex networks and for R will feature, however the subjects themselves correspond to advanced courses on social network analysis with the specialization on algebraic models and methods."--

Presented in a comprehensive manner, this book provides a comprehensive foundation in algebraic approaches for the analysis of different types of social networks such as multiple, signed, and affiliation networks. The study of such configurations corresponds to the structural analysis within the social sciences, and the methods applied for the analysis are in the areas of abstract algebra, combinatorics, and graph theory.

Current research in social networks has moved toward the examination of more realistic but also more complex social relations by which agents or actors are connected in multiple ways. Addressing this trend, this book offers hands-on training of the algebraic procedures presented along with the computer package multiplex, written by the book’s author specifically to perform analyses of multiple social networks. An introductory section on both complex networks and for R will feature, however the subjects themselves correspond to advanced courses on social network analysis with the specialization on algebraic models and methods.

List of Figures
xvii
List of Tables
xxi
Preface xxiii
Abbreviations xxv
Symbols xxvii
About the Companion Website xxxi
1 Structural Analysis with Algebra
1(24)
1.1 Preliminaries
1(1)
1.2 Graphs
2(3)
1.2.1 Graphs and Digraphs
2(1)
1.2.2 Multigraphs
3(1)
1.2.3 Signed Graph
3(1)
1.2.4 Bipartite Graph
4(1)
1.2.5 Valued Graph
4(1)
1.2.6 Multilevel Graph
5(1)
1.3 Matrices
5(3)
1.3.1 Affiliation Matrix
5(1)
1.3.2 Multiple Relations
6(1)
1.3.3 Incidence Matrix
6(1)
1.3.4 Valency Matrix
6(1)
1.3.5 Different Systems
7(1)
1.3.6 Graph and Matrix Representations
7(1)
1.4 Chains, Paths, and Other Graph Properties
8(1)
1.5 Algebra of Relations
9(1)
1.5.1 Generators and Compounds
9(1)
1.6 Operations on Social Networks
10(3)
1.6.1 Binary Operation on Relations
10(1)
1.6.2 Relational Composition
11(2)
1.7 Types and Properties of Relations
13(1)
1.8 Equivalence and Ordering
14(2)
1.8.1 Equivalence
14(1)
1.8.2 Partial Order
15(1)
1.8.3 Hierarchy
16(1)
1.9 Functions
16(3)
1.9.1 Identity and Empty Functions
18(1)
1.9.2 Transformations
19(1)
1.10 Homomorphism and Congruence
19(2)
1.10.1 Congruence Relations
20(1)
1.10.2 Kernel of a Homomorphism
20(1)
1.11 Structural Analysis with Algebra: Summary
21(1)
1.12 Learning Structural Analysis by Doing
22(3)
1.12.1 Getting Started
22(1)
1.12.2 Matrices
22(1)
1.12.3 Graphs
23(2)
2 Algebraic Structures
25(24)
2.1 Algebraic Structure Definition
25(1)
2.1.1 Closure
25(1)
2.2 Group Structure
26(4)
2.2.1 Cayley Graph
27(1)
2.2.2 Permutation Groups
28(1)
2.2.3 Presentation of Group Structures
29(1)
2.3 Group of Symmetries: Dihedral Groups
30(4)
2.3.1 Group of Symmetries of the Equilateral Triangle
30(2)
2.3.2 Group of Symmetries of the Square
32(2)
2.3.3 Generating Set in Symmetric Groups
34(1)
2.4 Semigroup
34(2)
2.4.1 Semigroup of Relations
35(1)
2.5 Semigroup and Group Properties
36(1)
2.5.1 Regular Elements
36(1)
2.5.2 Subsemigroups and Ideals
36(1)
2.6 Ring and Semiring
37(1)
2.6.1 Semiring
37(1)
2.7 Lattice Structure
38(3)
2.7.1 Congruence Lattice
39(1)
2.7.2 Modular and Distributive Lattice
40(1)
2.8 Algebraic Structures: Summary
41(1)
2.9 Learning Algebraic Structures by Doing
42(7)
2.9.1 Dihedral Group of the Equilateral Triangle D 3
42(2)
2.9.2 Dihedral Group of the Square D4
44(2)
2.9.3 Modular and Nonmodular Lattices
46(3)
3 Multiplex Network Configurations
49(34)
3.1 Multiple Networks
49(2)
3.1.1 Types of Multiple Networks
50(1)
3.2 Kinship Networks and Group Structure
51(2)
3.2.1 Marriage Types in Kinship Systems
52(1)
3.3 Rules for Marriage and Descent in the Kariera Society
53(3)
3.3.1 Group Structure and Set of Equations
55(1)
3.4 Algebraic Constraints
56(1)
3.5 Link Generalizations and Complex Structures
57(1)
3.6 Bundle Patterns
58(4)
3.6.1 Bundle Class Properties
59(1)
3.6.2 Bundle Isomorphic Classes
60(1)
3.6.3 Statistical Approach to Bundle Patterns
61(1)
3.7 Co-occurrence of Ties Model
62(2)
3.8 Relational Structure
64(4)
3.8.1 Strength of Weak Ties Model as Relational Structure
65(1)
3.8.2 Graph Representation of the Strength of Weak Ties
66(2)
3.9 Semigroup of Relations in Multiplex Networks
68(6)
3.9.1 Partial Order Relations and the Axiom of Quality
69(2)
3.9.2 Multiplication Table
71(3)
3.10 Partially Ordered Semigroup
74(2)
3.10.1 Partial Ordering in Xz
75(1)
3.11 Word and Edge Tables
76(1)
3.12 Multiplex Network Configurations: Summary
77(1)
3.13 Learning Multiplex Networks by Doing
78(5)
3.13.1 Kariera Kinship Network
78(1)
3.13.2 Multiplex Networks
79(1)
3.13.3 Strength of Weak Ties
80(1)
3.13.4 Relational Structure
80(3)
4 Positional Analysis and Role Structure
83(26)
4.1 Roles and Positions
83(1)
4.2 Network Homomorphism
84(3)
4.2.1 Weak and Strong Graph Homomorphisms
85(1)
4.2.2 Juncture Graph Homomorphism
86(1)
4.3 Global Equivalences
87(4)
4.3.1 Structural Equivalence
88(1)
4.3.2 Automorphic Equivalence
88(1)
4.3.3 Regular Equivalence
89(1)
4.3.4 Generalized Equivalence
90(1)
4.4 Global Equivalences Applied
91(3)
4.5 Local Equivalences
94(3)
4.5.1 Relation-Box R(W)
94(1)
4.5.2 Relation Plane and Role Relations in R(W)
95(1)
4.5.3 Local Role Equivalence
96(1)
4.6 Compositional Equivalence
97(2)
4.6.1 Formal Definition of Compositional Equivalence
98(1)
4.7 Positional Analysis with Compositional Equivalence
99(5)
4.7.1 Cumulated Person Hierarchy
99(2)
4.7.2 Set of Generators in Complex Networks
101(1)
4.7.3 Incorporating Actor Attributes
102(2)
4.8 Positional Analysis and Role Structure: Summary
104(1)
4.9 Learning Positional Analysis and Role Structure by Doing
105(4)
4.9.1 Equivalence Relations
105(4)
5 Role Structure in Multiplex Networks
109(36)
5.1 Directed Role Structures: Incubator Network A
110(9)
5.1.1 Social Positions in Network XA
111(1)
5.1.2 Modeling XA with Compositional Equivalence
112(2)
5.1.3 Cumulated Person Hierarchy HA
114(2)
5.1.4 Positional System lA
116(3)
5.2 Role Structure Incubator Network A
119(6)
5.2.1 Constructing Role Structures
120(1)
5.2.2 Particular Elements in the Role Structure
121(1)
5.2.3 Role Structure with Relational Contrast
122(3)
5.3 Undirected Role Structures: Florentine Families Network
125(7)
5.3.1 Positional Analysis of the Florentine Families Network
125(2)
5.3.2 Constructing Person Hierarchies, HF
127(2)
5.3.3 Family Attributes in XF
129(3)
5.4 Role Structure of the Florentine Families Network
132(5)
5.4.1 Interlock of Business, Marriage and Wealth Role Relations in DF
134(1)
5.4.2 Inclusion of Role Relations
135(2)
5.5 Role Structure in Multiplex Networks: Summary
137(1)
5.6 Learning Role Structure in Multiplex Networks by Doing
138(7)
5.6.1 Incubator Network A
138(1)
5.6.2 Florentine Families Network, XF
139(2)
5.6.3 Role Structure of XF with Wealth
141(4)
6 Decomposition of Role Structures
145(42)
6.1 Aggregation and Decomposition
145(2)
6.1.1 Homomorphic Reductions
147(1)
6.2 Synthesis Rules
147(2)
6.2.1 Direct Representation
147(1)
6.2.2 Subdirect Representation
148(1)
6.3 Lattice of Congruence Relations
149(1)
6.4 Factorization
150(2)
6.4.1 Atoms and their Meet-Complements
150(1)
6.4.2 Lattice of Homomorphisms of the Semigroup
151(1)
6.5 Congruences by Substitution Property
152(1)
6.6 Aggregation of Role Structures in QA
153(6)
6.6.1 Atoms with Meet-Complements in Role Structure QA
154(2)
6.6.2 Congruence Lattice Lπ(QA)
156(3)
6.7 Role Interlock of Incubator Network A
159(7)
6.7.1 Factorizing Set
159(5)
6.7.2 Hierarchy of Relations in QA
164(2)
6.8 Progressive Homomorphic Reduction of Factors in QA
166(3)
6.9 Role Structure for Incubator Network B
169(3)
6.9.1 Factorization of QB
169(1)
6.9.2 Congruence by Substitution Property in QB
170(2)
6.10 Role Interlock of Incubator Network C
172(1)
6.10.1 Decomposition of Qc
172(1)
6.11 Role Interlock of =QF for Florentine Families Network
173(4)
6.11.1 Congruence Classes in Role Structure QF
174(3)
6.12 Reduction Diagram
177(2)
6.13 Decomposition of Role Structures: Summary
179(1)
6.14 Learning Decomposition of Role Structures by Doing
180(7)
6.14.1 Factorization of Role Structure QA
180(3)
6.14.2 Decomposition of Florentine Families Role Structure QF
183(2)
6.14.3 Decomposition of Role Structure QB
185(2)
7 Signed Networks
187(28)
7.1 Structural Analysis of Signed Networks
187(1)
7.2 Social Influence Process
188(3)
7.2.7 Cohesion Influence
188(2)
7.2.2 Comparison and Influence
190(1)
7.3 Structural Balance
191(4)
7.3.7 Balance and Relational Composition
193(2)
7.4 Semirings for Structural Balance
195(4)
7.4.7 Valence Rules for Balance Semirings
196(3)
7.4.2 Closure Operations in Semirings
199(1)
7.5 Balance and Comparison Influence
199(2)
7.5.7 Weak Balanced Structures
201(1)
7.6 Looking for Structural Balance
201(8)
7.6.1 Balance Semiring in Signed Network Zσ
203(5)
7.6.2 Cluster Semiring in Signed Network XσA
208(1)
7.7 Signed Networks: Summary
209(1)
7.8 Learning Signed Networks by Doing
210(5)
7.8.1 Signed Structures in Figure 7.1
210(1)
7.8.2 Balance Semiring Structures in a Signed Triad
210(1)
7.8.3 Structural Balance in Incubator Network A, XA
211(1)
7.8.4 Balance Structures in Table 7.4
211(4)
8 Affiliation Networks
215(26)
8.1 Structural Analysis of Affiliation Networks
215(3)
8.1.1 Visualization and Partition of Two-mode Data
216(2)
7.2 Binomial Projection
218(2)
8.2 Common Affiliations
220(4)
8.2.1 Actors Perspective
220(2)
8.2.2 Events Perspective
222(1)
8.2.3 Affiliation Network with Bridge Organizations XBG20B
223(1)
8.3 Formal Concept Analysis
224(1)
8.4 Formal Concepts and Galois Derivations
225(3)
8.4.1 Concepts in the G20 Affiliation Network
226(2)
8.5 Concept Lattice and Ordering of Concepts
228(4)
8.8.7 Partial Ordering of the Concepts
228(1)
8.5.2 Concept Lattice of the Context
228(2)
8.5.3 Concept Lattice of Network Xb20
230(2)
8.6 Order Filters and Order Ideals
232(2)
8.6.7 Principal Order Filters
232(1)
8.6.2 Order Ideals and Principal Order Ideals
233(1)
8.7 Affiliation Networks: Summary
234(1)
8.8 Learning Affiliation Networks by Doing
235(6)
8.8.1 G20 Affiliation Network
235(1)
8.8.2 Bipartite Graphs in XBG20
235(1)
8.8.3 Co-affiliation Network of G20 Network
236(1)
8.8.4 Positional System of XBG20b with Events Classes
236(1)
8.8.5 Clustered Bipartite Graph and Binomial Projection of XBG20b
236(2)
8.8.6 Formal Concept Analysis
238(2)
8.8.7 Order Filters and Order Ideals
240(1)
9 Valued Networks
241(32)
9.1 Relational Structure of Valued Networks
242(7)
9.1.1 Valued Paths in the G20 Trade Network
242(3)
9.7.2 Constructing Valued Paths
245(1)
9.7.5 Semigroup and Equations of Valued Relations
246(1)
9.1.4 First Letter Law in Semigroup Structure
247(2)
9.2 Many-valued Contexts
249(6)
9.2.7 Conceptual Scaling
249(1)
9.2.2 Conceptual Scaling of XBG20
250(3)
9.2.3 Concept Lattices Concept lattices of Many-valued Contexts
253(2)
9.3 Pathfinder Network Analysis
255(3)
9.5.7 Pathfinder Semiring Pathfinder semiring
256(1)
9.3.2 Pathfinder Algorithm
257(1)
9.4 Pathfinder Semiring to Co-affiliation Network in XB20
258(1)
9.5 Triangle Inequality
259(3)
9.5.7 Application of Triangle Inequality triangle inequality to a Valued Configuration
260(1)
9.5.2 Triangle Inequality triangle inequality in Multiplex Networks
261(1)
9.6 Trade Network XVG20 with Triangle Inequality
262(2)
9.7 Valued Networks: Summary
264(1)
9.8 Learning Valued Networks by Doing
265(8)
9.8.1 Valued Network
265(1)
9.8.2 Semigroup of Valued Network with max-min Product
266(1)
9.8.3 Many-valued Contexts
267(2)
9.8.4 Pathfinder Semiring
269(2)
9.8.5 Triangle Inequality
271(2)
10 Multilevel Networks
273(28)
10.1 Structural Analysis of Multilevel Systems
273(1)
10.2 Visual Representation of Clients and Attorneys Multilevel Network
274(2)
10.2.1 Additional Features
276(1)
10.3 Multilevel Structure of the G20 Network
276(3)
10.3.1 Multilevel structure of all G20 countries MXG20
276(3)
10.4 Multilevel Positional System of G20 Network with Bridges
279(5)
10.4.1 Visual Interpretation of the Multilevel Structure in XMB20b
281(1)
10.4.2 Positional Analysis of XBG20b
282(1)
10.4.3 Depiction of Multilevel Positional System XMG20b
283(1)
10.5 Algebraic Approaches to Multilevel Networks
284(5)
10.5.1 G20 Multilevel Network
285(2)
10.5.2 Visualization of Multilevel Network Algebra
287(2)
10.5.3 Substantial Interpretation
289(1)
10.6 Reducing Complexity in XMG20b
289(2)
10.7 Further Algebraic Representations of Multilevel Structures
291(1)
10.8 Multilevel Networks: Summary
292(1)
10.9 Learning Multilevel Networks by Doing
293(8)
10.9.1 Multilevel Network `Clients and Attorneys'
293(1)
10.9.2 Multilevel Structure of G20 Network with Bridges
294(1)
10.9.3 Multilevel Structure ofG20 Trade and Affiliation Networks
295(1)
10.9.4 Positional System for the Algebraic Analysis
296(1)
10.9.5 Relational Structure of Multilevel Configurations
297(2)
10.9.6 Two-class Multilevel Positional System
299(2)
11 Comparing Relational Structures
301(34)
11.1 Comparing Structures with Algebraic Constraints
302(1)
11.2 Incubator Networks B and C
303(4)
11.2.1 Positional Analysis of XB and Xc
303(4)
11.3 Equality
307(3)
11.3.1 Set of Equations in Incubator Role Structures
307(3)
11.4 Hierarchy of Relations
310(2)
11.4.1 Set of Inclusions in Incubator Networks
311(1)
11.5 Shared Structure by Role Tables
312(4)
11.5.1 Lattice of Homomorphisms of the Semigroup
312(2)
11.5.2 Joint Homomorphic Reduction, JNTHOM
314(1)
11.5.3 Common Structure Semigroup, CSS
314(1)
11.5.4 What Constitutes a "Shared" Structure?
315(1)
11.6 Semigroup Tables with Joint Homomorphic Reduction
316(3)
11.6.1 JNTHOM of Aggregated Role tables QA
316(1)
11.6.2 JNTHOM of Aggregated Role tables QB and QC
317(1)
11.6.3 Joint Table for Incubator Networks
318(1)
11.7 Comparison Across Networks with Common Structure Semigroup
319(5)
11.7.1 CSS for Incubator networks A, B, and C
321(3)
11.7.2 CSS Order Role Structure for 2A-B-C
324(1)
11.8 Comparing Structures in Substantial Terms
324(3)
11.8.1 Hierarchy of Social Relations and Actor Attributes
324(3)
11.8.2 Set of Equations or Equality in QA, QB, and QC
327(1)
11.9 Structuring Effect of Role Relations in Incubators
327(2)
11.10 Comparing Relational Structures: Summary
329(1)
11.11 Learning Comparing Relational Structures by Doing
330(5)
11.11.1 Visualization of Incubator Networks B and C
330(1)
11.11.2 Positional Analysis and Role Structure for XB and Xc
330(1)
11.11.3 Decomposition of QB and Qc
331(2)
11.11.4 Equalities in Incubator Networks
333(2)
A Datasets
335(4)
Kariera kinship
335(1)
Incubators A, B, C
335(1)
Florentine families
336(1)
Clients and attorneys
336(1)
Group of twenty
336(3)
B Role structures of Incubator networks
339(8)
Role Structure of XA
339(1)
Role Structure of XB
339(1)
Positional system of Incubator network B
339(2)
Role tables in QB
341(1)
Role Structure of XC
342(1)
Positional system of Incubator network C
342(5)
C Valued data in G20 Trade network
347(6)
Group of Twenty Indicators
347(1)
Commodities in G20 Trade valued network
348(1)
Units of measure ofG20 country data
348(1)
G20 Trade valued network and salient structures
348(5)
D Layout visualization algorithms
353(18)
Force-directed
353(2)
Stress-majorization
355(3)
Laplacian Function
358(1)
New stress internal function
359(2)
E Role structure workflow
361(1)
Decomposition of Role structure 2B
361(1)
Incubator network B
361(1)
Positional analysis and Role structure
361(1)
Factorization
362(1)
Progressive factorization of Factors
363(7)
Aggregated structure of 2B
370(1)
Bibliography 371(6)
Index 377
J. ANTONIO RIVERO OSTOIC, PHD, is a post doctorate fellow at the School of Culture and Society, Aarhus University, Denmark, and a research associate at the University of San Simón (CESU). With a background in sociology and social sciences his research is mainly focused on social networks. He developed the R packages multiplex and multigraph for performing algebraic analysis and visualization of complex systems.
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