Chapter 1: Fundamentals of Graphs and Graph Representation Learning

This chapter covers the fundamentals of Fundamentals of Graphs and Graph Representation Learning, which fundamentals of graph theory. You will learn basic graph concepts (nodes, different types of graphs (trees, and Calculate graph features (degree.

Learning Objectives

By completing this chapter, you will master the following:

✅ Understand basic graph concepts (nodes, edges, directed/undirected graphs)
✅ Explain different types of graphs (trees, DAGs, complete graphs, bipartite graphs)
✅ Use different graph representations (adjacency matrix, adjacency list, edge list)
✅ Create and visualize graphs using NetworkX
✅ Calculate graph features (degree, clustering coefficient, centrality measures)
✅ Understand and implement the PageRank algorithm
✅ Understand random walk-based embedding methods (DeepWalk, Node2Vec)
✅ Implement node classification and link prediction using graph embeddings
✅ Apply community detection algorithms
✅ Perform social network analysis

1.1 Fundamentals of Graph Theory

What is a Graph?

A graph is a mathematical structure that represents relationships between objects. Many real-world problems can be represented as graphs, including social networks, molecular structures, road networks, and knowledge graphs.

“A graph $G$ is defined by a set of nodes (vertices) $V$ and a set of edges $E$: $G = (V, E)$“

Basic Terminology

Graph theory uses several fundamental concepts. A Node (Vertex) is a point representing an entity such as a person, web page, or atom. An Edge (Link) is a line representing a relationship between nodes, such as a friendship, hyperlink, or chemical bond. In a Directed Graph , edges have directionality like Twitter follow relationships, whereas in an Undirected Graph , edges have no directionality like Facebook friendships. A Weighted Graph assigns numerical weights to edges.

```mermaid
graph LR
    subgraph "Undirected Graph"
    A1((A)) --- B1((B))
    B1 --- C1((C))
    C1 --- A1
    A1 --- D1((D))
    end

    subgraph "Directed Graph"
    A2((A)) --> B2((B))
    B2 --> C2((C))
    C2 --> A2
    A2 --> D2((D))
    D2 --> B2
    end

    style A1 fill:#e3f2fd
    style B1 fill:#e3f2fd
    style C1 fill:#e3f2fd
    style D1 fill:#e3f2fd
    style A2 fill:#fff3e0
    style B2 fill:#fff3e0
    style C2 fill:#fff3e0
    style D2 fill:#fff3e0
```

Types of Graphs

Graph Type	Definition	Examples
Tree	Connected graph with no cycles	File system, organizational chart
DAG	Directed acyclic graph	Task dependencies, causal graphs
Complete Graph	Edges exist between all node pairs	Fully connected network
Bipartite Graph	Nodes can be divided into two groups	Recommendation system (user-item)
Cycle Graph	Forms a single cycle	Circular references, ring structure
Regular Graph	All nodes have equal degree	Crystal lattice, torus graph

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