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.
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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