Mathematics of Big Data Spreadsheets, Databases, Matrices, and Graphs

Preface
About the Authors
About the Cover
Acknowledgments
Applications and Practice
	Introduction and Overview
		Mathematics of Data
		Data in the World
		Mathematical Foundations
		Making Data Rigorous
		Conclusions, Exercises, and References
	Perspectives on Data
		Interrelations
		Spreadsheets
		Databases
		Matrices
		Graphs
		Map Reduce
		Other Perspectives
		Conclusions, Exercises, and References
	Dynamic Distributed Dimensional Data Model
		Background
		Design
		Matrix Mathematics
		Common SQL, NoSQL, NewSQL Interface
		Key-Value Store Database Schema
		Data-Independent Analytics
		Parallel Performance
		Computing on Masked Data
		Conclusions, Exercises, and References
	Associative Arrays and Musical Metadata
		Data and Metadata
		Dense Data
		Dense Operations
		Sparse Data
		Sparse Operations
		Conclusions, Exercises, and References
	Associative Arrays and Abstract Art
		Visual Abstraction
		Minimal Adjacency Array
		Symmetric Adjacency Array
		Weighted Adjacency Array
		Incidence Array
		Conclusions, Exercises, and References
	Manipulating Graphs with Matrices
		Introduction
		Matrix Indices and Values
		Composable Graph Operations and Linear Systems
		Matrix Graph Operations Overview
		Graph Algorithms and Diverse Semirings
		Conclusions, Exercises, and References
	Graph Analysis and Machine Learning Systems
		Introduction
		Data Representation
		Graph Construction
		Adjacency Array Graph Traversal
		Incidence Array Graph Traversal
		Vertex Degree Centrality
		Edge Degree Centrality
		Eigenvector Centrality
		Singular Value Decomposition
		PageRank
		Deep Neural Networks
		Conclusions, Exercises, and References
Mathematical Foundations
	Visualizing the Algebra of Associative Arrays
		Associative Array Analogs of Matrix Operations
		Abstract Algebra for Computer Scientists and Engineers
		Depicting Mathematics
		Associative Array Class Diagrams
		Set
		Semiring
		Linear Algebra
		Ordered Sets
		Boolean Algebra
		Associative Array Algebra
		Conclusions, Exercises, and References
	Defining the Algebra of Associative Arrays
		Operations on Sets
		Ordered Sets
		Supremum and Infimum
		Lattice
		The Semirings of Interest
		Conclusions, Exercises, and References
	Structural Properties of Associative Arrays
		Estimating Structure
		Associative Array Formal Definition
		Padding Associative Arrays with Zeros
		Zero, Null, Zero-Sum-Free
		Properties of Matrices and Associative Arrays
		Properties of Zero Padding
		Support and Size
		Image and Rank
		Example: Music
		Example: Art
		Properties of Element-Wise Addition
		Properties of Element-Wise Multiplication
		Array Multiplication
		Closure of Operations between Arrays
		Conclusions, Exercises, and References
	Graph Construction and Graphical Patterns
		Introduction
		Adjacency and Incidence Array Definitions
		Adjacency Array Construction
		Graph Construction with Different Semirings
		Special Arrays and Graphs
		Key Ordering
		Algebraic Properties
		Subobject Properties
		Conclusions, Exercises, and References
Linear Systems
	Survey of Common Transformations
		Array Transformations
		Identity
		Contraction
		Stretching
		Rotation
		Conclusions, Exercises, and References
	Maps and Bases
		Semimodules
		Linear Maps
		Linear Independence and Bases
		Existence of Bases
		Size of Bases
		Semialgebras and the Algebra of Arrays
		Conclusions, Exercises, and References
	Linearity of Associative Arrays
		The Null Space of Linear Maps
		Supremum-Blank Algebras
		Max-Blank Structure Theorem
		Examples of Supremum-Blank Algebras
		Explicit Computations of x(A,w) for Supremum-Blank Algebras
		Conclusions, Exercises, and References
	Eigenvalues and Eigenvectors
		Introduction
		Quasi-Inverses
		Existence of Eigenvalues for Idempotent Multiplication
		Strong Dependence and Characteristic Bipolynomial
		Eigenanalysis for Irreducible Matrices for Invertible Multiplication
		Eigen-Semimodules
		Singular Value Decomposition
		Conclusions, Exercises, and References
	Higher Dimensions
		d-Dimensional Associative Arrays
		Key Ordering and Two-Dimensional Projections
		Algebraic Properties
		Subarray Properties
		Conclusions, Exercises, and References
Appendix: Notation
Index
Answers to Selected Exercises