Network Science: A Mathematical and Computational Introduction to Graphs and Complex Systems using R

Master Network Science Through Mathematics, Computation, and R

Network Science: A Mathematical and Computational Introduction to Graphs and Complex Systems using R is a rigorous yet accessible textbook that teaches the mathematical foundations of graph theory and network analysis alongside hands-on R programming. From the structure of social networks to the mathematics of clustering and connectivity, every concept is developed from first principles and immediately reinforced with working R code - no external packages required.

Inspired by the Schaum's Outlines tradition of learning through worked examples, this book pairs formal definitions and proofs with step-by-step computations so readers build both theoretical understanding and practical skill simultaneously.

• Mathematical Foundations: Set theory, matrix algebra, and the graph-theoretic vocabulary underlying all of network science
• Network Representations: Edge lists, adjacency matrices, directed and undirected graphs, simple graphs, and multigraphs
• Node Degrees: Degree sequences, the Handshaking Lemma, in- and out-degree for directed networks
• Degree Distributions: Frequency tables, degree histograms, heavy-tailed distributions, and hub-heavy networks
• Paths and Walks: Walks, trails, paths, cycles, and matrix-power methods for counting indirect connections
• Triangles and V-Shapes: Local triangle counts, global triangle counts via tr(A3)/6, and open triads
• Network Clustering: Local clustering coefficient, global clustering coefficient, transitivity, and how clustering differs across network types

This textbook is written for undergraduate and graduate students in mathematics, statistics, computer science, and data science who want a mathematically grounded introduction to network analysis. It is equally well-suited for working professionals and researchers seeking a self-contained, example-driven reference. A background in introductory linear algebra and basic probability is helpful but not required - the book builds all necessary tools from scratch.

• Worked examples throughout: Every definition and theorem is followed by fully solved numerical examples, in the tradition of Schaum's Outlines
• Base R only: All computational examples use only base R - no igraph, no tidyverse, no external packages - so readers understand exactly what the code is doing
• Theory meets computation: Each mathematical result is paired with an R implementation that verifies the theory on concrete networks
• Clear visual diagrams: Network graphs, Venn diagrams, and degree histograms illustrate every major concept
• Self-contained: Mathematical prerequisites are introduced within the text, making the book accessible without a separate reference

Whether you are taking your first course in network science, teaching graph theory, or building a foundation for research in complex systems and data science, this book gives you the mathematical rigor and computational fluency to analyze real-world networks with confidence.