Graph Theory for Graph Neural Networks

Graph theory is a fundamental branch of mathematics that deals with the study of graphs and networks. A graph is a visual representation of complex data structures that helps us understand the relationships between different entities. Graph theory provides us with tools to model and analyze a vast array of real-world problems, such as transportation systems, social networks, and internet connectivity.

In this chapter, we will delve into the essentials of graph theory, covering three main topics: graph properties, graph concepts, and graph algorithms. We will begin by defining graphs and their components. We will then introduce the different types of graphs and explain their properties and applications. Next, we will cover fundamental graph concepts, objects, and measures, including the adjacency matrix. Finally, we will dive into graph algorithms, focusing on the two fundamental algorithms, breadth-first search (BFS) and depth-first search...

Technical requirements

All the code examples from this chapter can be found on GitHub at https://github.com/PacktPublishing/Hands-On-Graph-Neural-Networks-Using-Python/tree/main/Chapter02.

The installation steps required to run the code on your local machine can be found in the Preface of this book.

Introducing graph properties

In graph theory, a graph is a mathematical structure consisting of a set of objects, called vertices or nodes, and a set of connections, called edges, which link pairs of vertices. The notation is used to represent a graph, where is the graph, is the set of vertices, and is the set of edges.

The nodes of a graph can represent any objects, such as cities, people, web pages, or molecules, and the edges represent the relationships or connections between them, such as physical roads, social relationships, hyperlinks, or chemical bonds.

This section provides an overview of fundamental graph properties that will be used extensively in later chapters.

Directed graphs

One of the most basic properties of a graph is whether it is directed or undirected. In a directed graph, also called a digraph, each edge has a direction or orientation. This means that the edge connects two nodes in a particular direction, where one node is the source and the other...

Discovering graph concepts

In this section, we will explore some of the essential concepts in graph theory, including graph objects (such as degree and neighbors), graph measures (such as centrality and density), and the adjacency matrix representation.

Fundamental objects

One of the key concepts in graph theory is the degree of a node, which is the number of edges incident to this node. An edge is said to be incident on a node if that node is one of the edge’s endpoints. The degree of a node is often denoted by . It can be defined for both directed and undirected graphs:

• In an undirected graph, the degree of a vertex is the number of edges that are connected to it. Note that if the node is connected to itself (called a loop, or self-loop), it adds two to the degree.
• In a directed graph, the degree is divided into two types: indegree and outdegree. The indegree (denoted by ) of a node represents the number of edges that point towards that node, while the outdegree...

Exploring graph algorithms

Graph algorithms are critical in solving problems related to graphs, such as finding the shortest path between two nodes or detecting cycles. This section will discuss two graph traversal algorithms: BFS and DFS.

Breadth-first search

BFS is a graph traversal algorithm that starts at the root node and explores all the neighboring nodes at a particular level before moving to the next level of nodes. It works by maintaining a queue of nodes to visit and marking each visited node as it is added to the queue. The algorithm then dequeues the next node in the queue and explores all its neighbors, adding them to the queue if they haven’t been visited yet.

The behavior of a BFS is illustrated in Figure 2.7:

Figure 2.7 – Example of graph traversal made by a breadth-first search

Let’s now see how we can implement it in Python:

1. We create an empty graph and add edges with the add_edges_from() method:

   G...

Summary

In this chapter, we covered the essentials of graph theory, a branch of mathematics that studies graphs and networks. We began by defining what a graph is and explained the different types of graphs, such as directed, weighted, and connected graphs. We then introduced fundamental graph objects (including neighbors) and measures (such as centrality and density), which are used to understand and analyze graph structures.

Additionally, we discussed the adjacency matrix and its different representations. Finally, we explored the two fundamental graph algorithms, BFS and DFS, which form the foundation for developing more complex graph algorithms.

In Chapter 3, Creating Node Representations with DeepWalk, we will explore the DeepWalk architecture and its two components: Word2Vec and random walks. We will start by understanding the Word2Vec architecture and then implement it using a specialized library. Then, we will delve into the DeepWalk algorithm and implement random walks...