Algebraic Analysis of Multiple, Signed, and Affiliation Networks

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Title Page
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Contents
List of Figures
List of Tables
Preface
Abbreviations
Symbols
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Chapter 1 Structural Analysis with Algebra
	1.1 Preliminaries
	1.2 Graphs
		1.2.1 Graphs and Digraphs
		1.2.2 Multigraphs
		1.2.3 Signed Graph
		1.2.4 Bipartite Graph
		1.2.5 Valued Graph
		1.2.6 Multilevel Graph
	1.3 Matrices
		1.3.1 Affiliation Matrix
		1.3.2 Multiple Relations
		1.3.3 Incidence Matrix
		1.3.4 Valency Matrix
		1.3.5 Different Systems
		1.3.6 Graph and Matrix Representations
	1.4 Chains, Paths, and Other Graph Properties
	1.5 Algebra of Relations
		1.5.1 Generators and Compounds
	1.6 Operations on Social Networks
		1.6.1 Binary Operation on Relations
		1.6.2 Relational Composition
	1.7 Types and Properties of Relations
	1.8 Equivalence and Ordering
		1.8.1 Equivalence
		1.8.2 Partial Order
		1.8.3 Hierarchy
	1.9 Functions
		1.9.1 Identity and Empty Functions
		1.9.2 Transformations
	1.10 Homomorphism and Congruence
		1.10.1 Congruence Relations
		1.10.2 Kernel of a Homomorphism
	1.11 Structural Analysis with Algebra: Summary
	1.12 Learning Structural Analysis by Doing
		1.12.1 Getting Started
		1.12.2 Matrices
		1.12.3 Graphs
Chapter 2 Algebraic Structures
	2.1 Algebraic Structure Definition
		2.1.1 Closure
	2.2 Group Structure
		2.2.1 Cayley Graph
		2.2.2 Permutation Groups
		2.2.3 Presentation of Group Structures
	2.3 Group of Symmetries: Dihedral Groups
		2.3.1 Group of Symmetries of the Equilateral Triangle
		2.3.2 Group of Symmetries of the Square
		2.3.3 Generating Set in Symmetric Groups
	2.4 Semigroup
		2.4.1 Semigroup of Relations
	2.5 Semigroup and Group Properties
		2.5.1 Regular Elements
		2.5.2 Subsemigroups and Ideals
	2.6 Ring and Semiring
		2.6.1 Semiring
	2.7 Lattice Structure
		2.7.1 Congruence Lattice
		2.7.2 Modular and Distributive Lattice
	2.8 Algebraic Structures: Summary
	2.9 Learning Algebraic Structures by Doing
		2.9.1 Dihedral Group of the Equilateral Triangle D3
		2.9.2 Dihedral Group of the Square D4
		2.9.3 Modular and Nonmodular Lattices
Chapter 3 Multiplex Network Configurations
	3.1 Multiple Networks
		3.1.1 Types of Multiple Networks
	3.2 Kinship Networks and Group Structure
		3.2.1 Marriage Types in Kinship Systems
	3.3 Rules for Marriage and Descent in the Kariera Society
		3.3.1 Group Structure and Set of Equations
	3.4 Algebraic Constraints
	3.5 Link Generalizations and Complex Structures
	3.6 Bundle Patterns
		3.6.1 Bundle Class Properties
		3.6.2 Bundle Isomorphic Classes
		3.6.3 Statistical Approach to Bundle Patterns
	3.7 Co‐occurrence of Ties Model
	3.8 Relational Structure
		3.8.1 Strength of Weak Ties Model as Relational Structure
		3.8.2 Graph Representation of the Strength of Weak Ties
	3.9 Semigroup of Relations in Multiplex Networks
		3.9.1 Partial Order Relations and the Axiom of Quality
		3.9.2 Multiplication Table
	3.10 Partially Ordered Semigroup
		3.10.1 Partial Ordering in XZ
	3.11 Word and Edge Tables
	3.12 Multiplex Network Configurations: Summary
	3.13 Learning Multiplex Networks by Doing
		3.13.1 Kariera Kinship Network
		3.13.2 Multiplex Networks
		3.13.3 Strength of Weak Ties
		3.13.4 Relational Structure
Chapter 4 Positional Analysis and Role Structure
	4.1 Roles and Positions
	4.2 Network Homomorphism
		4.2.1 Weak and Strong Graph Homomorphisms
		4.2.2 Juncture Graph Homomorphism
	4.3 Global Equivalences
		4.3.1 Structural Equivalence
		4.3.2 Automorphic Equivalence
		4.3.3 Regular Equivalence
		4.3.4 Generalized Equivalence
	4.4 Global Equivalences Applied
	4.5 Local Equivalences
		4.5.1 Relation‐Box R(W)
		4.5.2 Relation Plane and Role Relations in R(W)
		4.5.3 Local Role Equivalence
	4.6 Compositional Equivalence
		4.6.1 Formal Definition of Compositional Equivalence
	4.7 Positional Analysis with Compositional Equivalence
		4.7.1 Cumulated Person Hierarchy, H
		4.7.2 Set of Generators in Complex Networks
		4.7.3 Incorporating Actor Attributes
	4.8 Positional Analysis and Role Structure: Summary
	4.9 Learning Positional Analysis and Role Structure by Doing
		4.9.1 Equivalence Relations
Chapter 5 Role Structure in Multiplex Networks
	5.1 Directed Role Structures: Incubator Network A
		5.1.1 Social Positions in Network XA
		5.1.2 Modeling XA with Compositional Equivalence
		5.1.3 Cumulated Person Hierarchy HA
		5.1.4 Positional System SA
	5.2 Role Structure Incubator Network A
		5.2.1 Constructing Role Structures
		5.2.2 Particular Elements in the Role Structure
		5.2.3 Role Structure with Relational Contrast
	5.3 Undirected Role Structures: Florentine Families Network
		5.3.1 Positional Analysis of the Florentine Families Network
		5.3.2 Constructing Person Hierarchies, HF
		5.3.3 Family Attributes in XF
	5.4 Role Structure of the Florentine Families Network
		5.4.1 Interlock of Business, Marriage and Wealth Role Relations in QF
		5.4.2 Inclusion of Role Relations
	5.5 Role Structure in Multiplex Networks: Summary
	5.6 Learning Role Structure in Multiplex Networks by Doing
		5.6.1 Incubator Network A
		5.6.2 Florentine Families Network, XF
		5.6.3 Role Structure of XF with Wealth
Chapter 6 Decomposition of Role Structures
	6.1 Aggregation and Decomposition
		6.1.1 Homomorphic Reductions
	6.2 Synthesis Rules
		6.2.1 Direct Representation
		6.2.2 Subdirect Representation
	6.3 Lattice of Congruence Relations
	6.4 Factorization
		6.4.1 Atoms and their Meet‐Complements
		6.4.2 Lattice of Homomorphisms of the Semigroup
	6.5 Congruences by Substitution Property
	6.6 Aggregation of Role Structures in QA
		6.6.1 Atoms with Meet‐Complements in Role Structure QA
		6.6.2 Congruence Lattice Lπ(QA)
	6.7 Role Interlock of Incubator Network A
		6.7.1 Factorizing Set
		6.7.2 Hierarchy of Relations in QA
	6.8 Progressive Homomorphic Reduction of Factors in QA
	6.9 Role Structure for Incubator Network B
		6.9.1 Factorization of QB
		6.9.2 Congruence by Substitution Property in QB
	6.10 Role Interlock of Incubator Network C
		6.10.1 Decomposition of QC
	6.11 Role Interlock of QF for Florentine Families Network
		6.11.1 Congruence Classes in Role Structure QF
	6.12 Reduction Diagram
	6.13 Decomposition of Role Structures: Summary
	6.14 Learning Decomposition of Role Structures by Doing
		6.14.1 Factorization of Role Structure QA
		6.14.2 Decomposition of Florentine Families Role Structure QF
		6.14.3 Decomposition of Role Structure QB
Chapter 7 Signed Networks
	7.1 Structural Analysis of Signed Networks
	7.2 Social Influence Process
		7.2.1 Cohesion Influence
		7.2.2 Comparison and Influence
	7.3 Structural Balance
		7.3.1 Balance and Relational Composition
	7.4 Semirings for Structural Balance
		7.4.1 Valence Rules for Balance Semirings
		7.4.2 Closure Operations in Semirings
	7.5 Balance and Comparison Influence
		7.5.1 Weak Balanced Structures
	7.6 Looking for Structural Balance
		7.6.1 Balance Semiring in Signed Network XAσ
		7.6.2 Cluster Semiring in Signed Network XAσ
	7.7 Signed Networks: Summary
	7.8 Learning Signed Networks by Doing
		7.8.1 Signed Structures in Figure 7.1
		7.8.2 Balance Semiring Structures in a Signed Triad
		7.8.3 Structural Balance in Incubator Network A, XA
		7.8.4 Balance Structures in Table 7.4
Chapter 8 Affiliation Networks
	8.1 Structural Analysis of Affiliation Networks
		8.1.1 Visualization and Partition of Two‐mode Data
		8.1.2 Binomial Projection
	8.2 Common Affiliations
		8.2.1 Actors Perspective
		8.2.2 Events Perspective
		8.2.3 Affiliation Network with Bridge Organizations XG20bB
	8.3 Formal Concept Analysis
	8.4 Formal Concepts and Galois Derivations
		8.4.1 Concepts in the G20 Affiliation Network
	8.5 Concept Lattice and Ordering of Concepts
		8.5.1 Partial Ordering of the Concepts
		8.5.2 Concept Lattice of the Context
		8.5.3 Concept Lattice of Network XG20B
	8.6 Order Filters and Order Ideals
		8.6.1 Principal Order Filters
		8.6.2 Order Ideals and Principal Order Ideals
	8.7 Affiliation Networks: Summary
	8.8 Learning Affiliation Networks by Doing
		8.8.1 G20 Affiliation Network
		8.8.2 Bipartite Graphs in XG20B
		8.8.3 Co‐affiliation Network of G20 Network
		8.8.4 Positional System of XG20bB with Events Classes
		8.8.5 Clustered Bipartite Graph and Binomial Projection of XG20bB
		8.8.6 Formal Concept Analysis
		8.8.7 Order Filters and Order Ideals
Chapter 9 Valued Networks
	9.1 Relational Structure of Valued Networks
		9.1.1 Valued Paths in the G20 Trade Network
		9.1.2 Constructing Valued Paths
		9.1.3 Semigroup and Equations of Valued Relations
		9.1.4 First Letter Law in Semigroup Structure
	9.2 Many‐valued Contexts
		9.2.1 Conceptual Scaling
		9.2.2 Conceptual Scaling of XG20B
		9.2.3 Concept Lattices of Many‐valued Contexts
	9.3 Pathfinder Network Analysis
		9.3.1 Pathfinder Semiring
		9.3.2 Pathfinder Algorithm
	9.4 Pathfinder Semiring to Co‐affiliation Network in XG20B
	9.5 Triangle Inequality
		9.5.1 Application of Triangle Inequality to a Valued Configuration
		9.5.2 Triangle Inequality in Multiplex Networks
	9.6 Trade Network XG20V with Triangle Inequality
	9.7 Valued Networks: Summary
	9.8 Learning Valued Networks by Doing
		9.8.1 Valued Network
		9.8.2 Semigroup of Valued Network with max‐min Product
		9.8.3 Many‐valued Contexts
		9.8.4 Pathfinder Semiring
		9.8.5 Triangle Inequality
Chapter 10 Multilevel Networks
	10.1 Structural Analysis of Multilevel Systems
	10.2 Visual Representation of Clients and Attorneys Multilevel Network
		10.2.1 Additional Features
	10.3 Multilevel Structure of the G20 Network
		10.3.1 Multilevel structure of all G20 countries XG20M
	10.4 Multilevel Positional System of G20 Network with Bridges
		10.4.1 Visual Interpretation of the Multilevel Structure in XG20bB
		10.4.2 Positional Analysis of XG20bM
		10.4.3 Depiction of Multilevel Positional System SG20bM
	10.5 Algebraic Approaches to Multilevel Networks
		10.5.1 G20 Multilevel Network
		10.5.2 Visualization of Multilevel Network Algebra
		10.5.3 Substantial Interpretation
	10.6 Reducing Complexity in XG20bM
	10.7 Further Algebraic Representations of Multilevel Structures
	10.8 Multilevel Networks: Summary
	10.9 Learning Multilevel Networks by Doing
		10.9.1 Multilevel Network ‘Clients and Attorneys’
		10.9.2 Multilevel Structure of G20 Network with Bridges
		10.9.3 Multilevel Structure of G20 Trade and Affiliation Networks
		10.9.4 Positional System for the Algebraic Analysis
		10.9.5 Relational Structure of Multilevel Configurations
		10.9.6 Two‐class Multilevel Positional System
Chapter 11 Comparing Relational Structures
	11.1 Comparing Structures with Algebraic Constraints
	11.2 Incubator Networks B and C
		11.2.1 Positional Analysis of XB and XC
	11.3 Equality
		11.3.1 Set of Equations in Incubator Role Structures
	11.4 Hierarchy of Relations
		11.4.1 Set of Inclusions in Incubator Networks
	11.5 Shared Structure by Role Tables
		11.5.1 Lattice of Homomorphisms of the Semigroup
		11.5.2 Joint Homomorphic Reduction, JNTHOM
		11.5.3 Common Structure Semigroup, CSS
		11.5.4 What Constitutes a “Shared” Structure?
	11.6 Semigroup Tables with Joint Homomorphic Reduction
		11.6.1 JNTHOM of Aggregated Role tables QA
		11.6.2 JNTHOM of Aggregated Role tables QB and QC
		11.6.3 Joint Table for Incubator Networks
	11.7 Comparison Across Networks with Common Structure Semigroup
		11.7.1 CSS for Incubator networks A, B, and C
		11.7.2 CSS Order Role Structure for QA−B−C
	11.8 Comparing Structures in Substantial Terms
		11.8.1 Hierarchy of Social Relations and Actor Attributes
		11.8.2 Set of Equations or Equality in QA, QB, and QC
	11.9 Structuring Effect of Role Relations in Incubators
	11.10 Comparing Relational Structures: Summary
	11.11 Learning Comparing Relational Structures by Doing
		11.11.1 Visualization of Incubator Networks B and C
		11.11.2 Positional Analysis and Role Structure for XB and XC
		11.11.3 Decomposition of QB and QC
		11.11.4 Equalities in Incubator Networks
Appendix A Datasets
Appendix B Role structures of Incubator networks
Appendix C Valued data in G20 Trade network
Appendix D Layout visualization algorithms
Appendix E Role structure workflow
INDEX
EULA