---
title: What Is Graph Theory for Neural Networks?
description: Explore the fundamentals of graph theory in neural networks, including concepts like nodes, edges, and message passing in graph neural networks for various applications.
image: https://blog.rheinwerk-computing.com/hubfs/What%20Is%20Graph%20Theory%20for%20Neural%20Networks_.jpg
---
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# What Is Graph Theory for Neural Networks?
 by [Rheinwerk Computing](https://blog.rheinwerk-computing.com/author/rheinwerk-computing) on October 21, 2026
A *graph* is a mathematical structure that represents a set of objects and the relationships among those objects. A graph always consists of two basic components:
- **Nodes:** These are the objects in the graph. For example, the nodes in a social network can be people.
- **Edges:** These represent the connections between or among nodes. In a social network, an edge would represent a friendly relationship between two people.
There are two types of edges: directional and nondirectional. Directional edges have a direction, like a one-way street. In the example of the social network, a directional edge can reflect the interest of person A in person B. If person A likes person B and person B does not like person A, then that’s an example of a directional edge. If the two people like each other, then that’s an example of a nondirectional or bidirectional edge.
Graphs are used to model complex systems in many areas. In addition to social networks, they can model biological networks, transportation routes, and citations between scientific papers.
## Graphs and the Adjacency Matrix
This figure shows a simple network structure with five nodes (A to E), which are connected to one another.
How can we model the connections between the nodes? This is quite simple with the help of an *adjacency matrix*, which is a square matrix that represents the connections in a graph. The matrix is structured in such a way that both the rows and the columns represent the nodes. The rows can represent the source node, and the columns can represent the target node. The values in the matrix indicate whether there is a connection between two nodes.
The figure below represents the adjacency matrix for the graph shown above. For example, a connection points from node B to node C, and we can read this connection in the matrix by searching in the second row (corresponding to the source node B) for the column with the target node (the third column for node C). At the intersection of both, there is the value 1, which stands for the directed connection. The following list shows how to create the entries in the matrix.
- An entry of means that there is an edge between node i and node j.
- If the edge is undirected, automatically applies as well.
- An entry of means that there is no edge between i and j.
For an undirected graph, the adjacency matrix is always symmetrical since the connection between A and B is the same as the connection between B and A. But for directed graphs, the matrix is asymmetrical, as shown in the example.
## Features
The nodes can also contain certain characteristics. If the nodes represent people, then the characteristics can represent specific properties of those people. The figure below shows an example of a feature matrix for the five nodes.
So how can we train a graph neural network? To do it, we need to understand the concept of passing messages.
## Passing Messages
The starting point, as shown in the figure below, is a graph that consists of five nodes. A GNN processes the information from the graph to generate a new representation for the target node (B).
This process takes place in the following steps:
1. The direct neighbors of B are identified. In this case, these are nodes C, D, and A.
2. The *GNN layer* collects the properties (known as *messages*) of these neighboring nodes, which are the feature vectors. Each of the brown boxes in the figure represents an aggregation function that processes the features of the neighbors, and this function can be an average, a sum, or a more complex operation.
3. The aggregated information from the neighbors is combined with the original features of the target node (B). This combination takes place in the dark blue box, and the result is a new, updated feature representation for node B, which now contains knowledge about its neighbors.
This process is repeated for each node in the graph. The information flows through several layers of a GNN over longer distances, so that a node incorporates the features of its “neighbors of the neighbors” and so on. This creates a representation (known as an *embedding*) that reflects not only the local structure but also the more global structures of the graph.
With each GNN layer, the embedding of a node becomes richer as it picks up information from increasingly distant neighbors. At the end of the process, the final embeddings have the entire graph structure encoded in them, which makes them very useful for downstream tasks such as classification.
## Use Cases
GNNs are very special neural networks that are ideal for special applications.
The above figure shows some of their most common use cases, as follows:
### Node Classification
The goal of node classification is to predict a category or characteristic for a single node in a graph. The GNN uses not only the characteristics of the respective node but also the information on its direct neighbors to obtain a more accurate classification. We can use node classification to detect fraudsters or spam bots in social networks because the GNN analyzes their connection patterns to other suspicious accounts. Similarly, we can use node classification to detect conspicuous or fraudulent accounts in a network of money flows.
### Link Prediction
Link prediction aims to predict the probability of a connection between two nodes that doesn’t exist in the graph. The model learns the patterns that lead to existing connections and then applies this knowledge to new pairs. One such use case is in recommendation systems. A GNN can predict whether a user might buy a product by analyzing connections among similar users and their purchasing decisions. We can use link prediction to generate friendship and partner suggestions on social networks or dating platforms, and link prediction is also used extensively in knowledge graphs to complete missing facts (e.g., to predict the country of origin of a particular company), thereby enriching and validating large databases. It’s also indispensable in biological networks, for example, for predicting new protein-protein interactions in genome research.
### Graph Classification
Graph classification involves assigning a category to the entire graph, not just to individual nodes. The GNN aggregates the information from all nodes and their connections to create a global representation that characterizes the entire graph. Graph classification is very commonly used in chemistry to represent molecules as graphs, with atoms as nodes and chemical bonds as connections between atoms. A GNN can classify the entire molecular graph to then predict, for example, whether the molecule is toxic or has a certain property.
## Conclusion
Graph neural networks are worth considering whenever the connections in your data carry as much meaning as the data points themselves. Once a problem can be expressed as nodes, edges, and features, the same message-passing approach applies whether you’re flagging a suspicious account, suggesting a product, or screening a molecule for toxicity. The harder work usually comes before any training starts, in deciding what your nodes and connections should actually represent. Get that model of the graph right, and the GNN has plenty to work with.
Editor’s note: This post has been adapted from a section of the book [*Generative AI with Python: The Developer’s Guide to Pretrained LLMs, Vector Databases, Retrieval-Augmented Generation, and Agentic Systems*](https://www.amazon.com/Generative-Python-Developers-Pretrained-Generation/dp/1493226908?maas=maas_adg_65C1F2AB72EBBA6CEA13210F2B44DDD3_afap_abs&ref_=aa_maas&tag=maas) by [Bert Gollnick](https://www.linkedin.com/in/bertgollnick/). Bert is a senior data scientist who specializes in renewable energies. For many years, he has taught courses about data science and machine learning, and more recently, about generative AI and natural language processing. Bert studied aeronautics at the Technical University of Berlin and economics at the University of Hagen. His main areas of interest are machine learning and data science.
## Recommendation
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