Deep Learning from Scratch Series: Graph Neural Networks with TensorFlow

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This blog teaches you how to build and train a graph neural network with TensorFlow and apply it to a graph analysis problem step by step.

1. Introduction

In this blog, you will learn how to build and train a graph neural network (GNN) with TensorFlow and apply it to a graph analysis problem. Graph neural networks are a type of deep learning model that can operate on graph-structured data, such as social networks, citation networks, molecular graphs, and knowledge graphs. Graph neural networks can learn from the features and relations of the nodes and edges in a graph, and produce node-level, edge-level, or graph-level outputs.

Graph neural networks are a powerful and versatile tool for graph analysis, as they can perform tasks such as node classification, link prediction, graph clustering, graph generation, and graph representation learning. In this blog, you will focus on the node classification task, which aims to assign a label to each node in a graph based on its features and neighborhood. For example, you can use node classification to predict the role of a user in a social network, the topic of a paper in a citation network, or the function of a protein in a molecular graph.

To implement a graph neural network with TensorFlow, you will use the Neural Structured Learning (NSL) framework, which provides tools and libraries for learning with structured data. NSL supports various types of graph neural networks, such as graph convolutional networks, graph attention networks, and graph sage networks. In this blog, you will use the graph convolutional network (GCN) model, which is one of the most popular and widely used graph neural network architectures. A GCN consists of multiple layers of graph convolution, which is a generalization of the standard convolution operation to graph-structured data.

By the end of this blog, you will be able to:

  • Understand the basics of graphs and graph neural networks
  • Implement a graph convolution layer with TensorFlow
  • Build and train a graph neural network model with NSL
  • Evaluate and visualize the model performance on a graph dataset

Are you ready to dive into the world of graph neural networks? Let’s get started!

2. What are Graphs and Graph Neural Networks?

Before you can implement a graph neural network (GNN) with TensorFlow, you need to understand what graphs and graph neural networks are, and why they are useful for graph analysis. In this section, you will learn the basics of graphs and graph neural networks, and see some examples of their applications.

A graph is a data structure that consists of a set of nodes (or vertices) and a set of edges (or links) that connect the nodes. Nodes can represent any entity, such as a person, a document, a molecule, or a concept. Edges can represent any relation, such as a friendship, a citation, a bond, or a similarity. Nodes and edges can also have features, such as attributes, labels, or weights, that describe their properties or characteristics.

Graphs are a natural way to represent many types of data that have complex and irregular structures, such as social networks, citation networks, molecular graphs, and knowledge graphs. Graphs can capture the rich and diverse information that is embedded in these data, such as the features and relations of the entities, the patterns and dynamics of the interactions, and the structure and function of the systems.

However, graphs are also challenging to analyze, as they do not have a fixed or regular shape, and they can vary in size, density, and complexity. Traditional machine learning methods, such as linear models, neural networks, or convolutional networks, are not well suited for graph data, as they assume that the data are independent and identically distributed (i.i.d.), and that they have a fixed or regular dimensionality or topology. Therefore, new methods are needed to handle graph data effectively and efficiently.

This is where graph neural networks come in. Graph neural networks are a type of deep learning model that can operate on graph-structured data, and learn from the features and relations of the nodes and edges in a graph. Graph neural networks can produce node-level, edge-level, or graph-level outputs, depending on the task and the objective. For example, a graph neural network can classify the nodes in a graph, predict the links between the nodes, cluster the nodes into groups, generate new graphs, or learn a representation of the whole graph.

Graph neural networks are a powerful and versatile tool for graph analysis, as they can perform various tasks that are relevant and important for many domains and applications, such as:

  • Node classification: Assigning a label to each node in a graph based on its features and neighborhood. For example, predicting the role of a user in a social network, the topic of a paper in a citation network, or the function of a protein in a molecular graph.
  • Link prediction: Estimating the probability of the existence or the weight of an edge between two nodes in a graph based on their features and neighborhood. For example, recommending friends or products in a social network, finding missing citations in a citation network, or discovering new drug-target interactions in a molecular graph.
  • Graph clustering: Partitioning the nodes in a graph into groups based on their features and relations. For example, detecting communities or anomalies in a social network, identifying research areas or trends in a citation network, or finding functional modules or pathways in a molecular graph.
  • Graph generation: Creating new graphs that match some given criteria or specifications. For example, synthesizing realistic social networks, citation networks, or molecular graphs.
  • Graph representation learning: Learning a low-dimensional vector representation of a graph that preserves its structural and semantic properties. For example, embedding a social network, a citation network, or a molecular graph into a vector space that can be used for downstream tasks such as classification, clustering, or visualization.

Now that you have a basic understanding of graphs and graph neural networks, you are ready to implement a graph neural network with TensorFlow. In the next section, you will learn how to install and import TensorFlow and other libraries that you will need for this tutorial.

2.1. Graphs and their properties

In this section, you will learn more about graphs and their properties, and how they can be represented and manipulated with Python. You will also see some examples of graph datasets that you can use for your graph analysis tasks.

A graph is a data structure that consists of a set of nodes (or vertices) and a set of edges (or links) that connect the nodes. Nodes can represent any entity, such as a person, a document, a molecule, or a concept. Edges can represent any relation, such as a friendship, a citation, a bond, or a similarity. Nodes and edges can also have features, such as attributes, labels, or weights, that describe their properties or characteristics.

There are different types of graphs, depending on the nature and direction of the edges, and the number and structure of the nodes. Some common types of graphs are:

  • Undirected graph: A graph where the edges have no direction, meaning that the relation between the nodes is symmetric. For example, a social network where the edges represent mutual friendships.
  • Directed graph: A graph where the edges have a direction, meaning that the relation between the nodes is asymmetric. For example, a citation network where the edges represent one-way citations.
  • Weighted graph: A graph where the edges have a numerical value, meaning that the relation between the nodes has a magnitude or strength. For example, a transportation network where the edges represent the distance or cost between the nodes.
  • Multi-graph: A graph where there can be multiple edges between the same pair of nodes, meaning that the relation between the nodes can have different types or modes. For example, a communication network where the edges represent different channels or mediums between the nodes.
  • Hypergraph: A graph where the edges can connect more than two nodes, meaning that the relation between the nodes can be higher-order or complex. For example, a knowledge graph where the edges represent logical or semantic relations between the nodes.

Graphs can be represented and manipulated with Python using various libraries and packages, such as NetworkX, graph-tool, or igraph. These libraries provide various functions and methods to create, modify, analyze, and visualize graphs. For example, you can use NetworkX to create a simple undirected graph with four nodes and five edges as follows:

import networkx as nx # import NetworkX library
G = nx.Graph() # create an empty undirected graph
G.add_nodes_from([1, 2, 3, 4]) # add four nodes with labels 1, 2, 3, and 4
G.add_edges_from([(1, 2), (1, 3), (1, 4), (2, 3), (3, 4)]) # add five edges between the nodes
nx.draw(G, with_labels=True) # draw the graph with node labels

There are many graph datasets that you can use for your graph analysis tasks, such as social networks, citation networks, molecular graphs, and knowledge graphs. Some examples of graph datasets are:

  • Stanford Network Analysis Project (SNAP): A collection of large-scale network datasets from various domains, such as social networks, web graphs, communication networks, collaboration networks, and citation networks.
  • Open Graph Benchmark (OGB): A collection of realistic and challenging graph datasets from various domains, such as molecular property prediction, protein function prediction, and recommendation systems.
  • Kaggle Graph Datasets: A collection of graph datasets from various domains, such as social networks, biological networks, transportation networks, and text networks.

Now that you have learned more about graphs and their properties, you are ready to learn about graph neural networks and their applications. In the next section, you will see how graph neural networks can operate on graph-structured data and perform various graph analysis tasks.

2.2. Graph Neural Networks and their applications

In this section, you will learn more about graph neural networks and their applications, and how they can operate on graph-structured data and perform various graph analysis tasks. You will also see some examples of graph neural network architectures and models that you can use for your graph analysis tasks.

A graph neural network (GNN) is a type of deep learning model that can operate on graph-structured data, and learn from the features and relations of the nodes and edges in a graph. A GNN consists of two main components: a graph encoder and a graph decoder. The graph encoder takes a graph as input and produces a latent representation of the graph, which captures its structural and semantic properties. The graph decoder takes the latent representation of the graph and produces an output, which can be node-level, edge-level, or graph-level, depending on the task and the objective.

The graph encoder is the core component of a GNN, as it defines how the GNN processes and aggregates the information from the graph. The graph encoder typically consists of multiple layers of graph convolution, which is a generalization of the standard convolution operation to graph-structured data. Graph convolution can be seen as a message passing scheme, where each node in the graph receives messages from its neighbors, and updates its representation based on the messages and its own features. Graph convolution can capture the local and global structure of the graph, as well as the features and relations of the nodes and edges.

There are different types of graph convolution, depending on how the messages are computed and aggregated, and how the node representations are updated. Some common types of graph convolution are:

  • Graph Convolutional Network (GCN): A type of graph convolution that computes the messages as a linear transformation of the node features, and aggregates them as a weighted sum based on the degree of the nodes. The node representations are updated as a non-linear activation of the aggregated messages and the node features. GCN is one of the most popular and widely used graph convolution methods, as it is simple and effective.
  • Graph Attention Network (GAT): A type of graph convolution that computes the messages as a linear transformation of the node features, and aggregates them as a weighted sum based on the attention scores of the nodes. The attention scores are learned by a neural network that measures the similarity between the nodes. The node representations are updated as a non-linear activation of the aggregated messages and the node features. GAT is a powerful and flexible graph convolution method, as it can learn the importance of the nodes and their relations.
  • GraphSAGE: A type of graph convolution that computes the messages as a concatenation of the node features and the neighbor features, and aggregates them as a mean, max, or LSTM pooling. The node representations are updated as a linear transformation of the aggregated messages and the node features. GraphSAGE is an efficient and scalable graph convolution method, as it can handle large and dynamic graphs.

The graph decoder is the component of a GNN that defines how the GNN produces the output from the latent representation of the graph. The graph decoder can be customized for different tasks and objectives, such as node classification, link prediction, graph clustering, graph generation, or graph representation learning. For example, for node classification, the graph decoder can be a softmax layer that predicts the label of each node based on its representation. For link prediction, the graph decoder can be a sigmoid layer that predicts the probability of an edge between two nodes based on their representations. For graph clustering, the graph decoder can be a k-means layer that assigns each node to a cluster based on its representation. For graph generation, the graph decoder can be a recurrent neural network that generates new nodes and edges based on the representation of the existing graph. For graph representation learning, the graph decoder can be a contrastive learning layer that learns a low-dimensional vector representation of the graph that preserves its structural and semantic properties.

Graph neural networks are a powerful and versatile tool for graph analysis, as they can perform various tasks that are relevant and important for many domains and applications, such as social networks, citation networks, molecular graphs, and knowledge graphs. Some examples of graph neural network architectures and models that you can use for your graph analysis tasks are:

  • GCN: Graph Convolutional Networks: A graph neural network model that uses GCN as the graph encoder and softmax as the graph decoder for semi-supervised node classification.
  • GAT: Graph Attention Networks: A graph neural network model that uses GAT as the graph encoder and softmax as the graph decoder for semi-supervised node classification.
  • GraphSAGE: Inductive Representation Learning on Large Graphs: A graph neural network model that uses GraphSAGE as the graph encoder and softmax as the graph decoder for inductive node classification.
  • Graph Nets: A graph neural network framework that provides various graph encoder and graph decoder modules for building graph neural network models for different tasks, such as graph classification, graph regression, graph generation, and graph manipulation.
  • OGB: Open Graph Benchmark: A graph neural network benchmark that provides various graph datasets and graph neural network models for different tasks, such as molecular property prediction, protein function prediction, and recommendation systems.

Now that you have learned more about graph neural networks and their applications, you are ready to implement a graph neural network with TensorFlow. In the next section, you will learn how to install and import TensorFlow and other libraries that you will need for this tutorial.

3. How to Implement a Graph Neural Network with TensorFlow

In this section, you will learn how to implement a graph neural network (GNN) with TensorFlow, using the Neural Structured Learning (NSL) framework. You will use the graph convolutional network (GCN) model, which is one of the most popular and widely used graph neural network architectures. You will apply the GCN model to the node classification task, which aims to assign a label to each node in a graph based on its features and neighborhood. You will use the Cora dataset, which is a citation network of scientific papers, where the nodes are papers, the edges are citations, and the labels are research topics.

To implement a graph neural network with TensorFlow, you will need to follow these steps:

  1. Install and import TensorFlow and other libraries
  2. Loading and preprocessing a graph dataset
  3. Defining and creating a graph convolution layer
  4. Building and training a graph neural network model
  5. Evaluating and visualizing the model performance

In this section, you will focus on the first step, which is installing and importing TensorFlow and other libraries. You will need to install and import the following libraries:

  • TensorFlow: A machine learning platform that provides various tools and libraries for building and deploying deep learning models. You will use TensorFlow to create and train your graph neural network model.
  • Neural Structured Learning: A TensorFlow framework that provides tools and libraries for learning with structured data, such as graphs. You will use NSL to implement the graph convolution layer and the graph neural network model.
  • NetworkX: A Python library for creating, manipulating, and analyzing graphs. You will use NetworkX to load and preprocess the graph dataset.
  • Matplotlib: A Python library for creating and displaying plots and figures. You will use Matplotlib to visualize the graph dataset and the model performance.

To install TensorFlow and NSL, you can use the pip command as follows:

pip install --upgrade tensorflow
pip install neural-structured-learning

To install NetworkX and Matplotlib, you can use the pip command as follows:

pip install networkx
pip install matplotlib

To import TensorFlow and NSL, you can use the import statement as follows:

import tensorflow as tf # import TensorFlow library
import neural_structured_learning as nsl # import Neural Structured Learning library

To import NetworkX and Matplotlib, you can use the import statement as follows:

import networkx as nx # import NetworkX library
import matplotlib.pyplot as plt # import Matplotlib library

Now that you have installed and imported TensorFlow and other libraries, you are ready to load and preprocess the graph dataset. In the next section, you will learn how to load and preprocess the Cora dataset, which is a citation network of scientific papers.

3.1. Installing and importing TensorFlow and other libraries

In this section, you will learn how to install and import TensorFlow and other libraries that you will need to implement a graph neural network with TensorFlow. You will need to install and import the following libraries:

  • TensorFlow: A machine learning platform that provides various tools and libraries for building and deploying deep learning models. You will use TensorFlow to create and train your graph neural network model.
  • Neural Structured Learning: A TensorFlow framework that provides tools and libraries for learning with structured data, such as graphs. You will use NSL to implement the graph convolution layer and the graph neural network model.
  • NetworkX: A Python library for creating, manipulating, and analyzing graphs. You will use NetworkX to load and preprocess the graph dataset.
  • Matplotlib: A Python library for creating and displaying plots and figures. You will use Matplotlib to visualize the graph dataset and the model performance.

To install TensorFlow and NSL, you can use the pip command as follows:

pip install --upgrade tensorflow
pip install neural-structured-learning

To install NetworkX and Matplotlib, you can use the pip command as follows:

pip install networkx
pip install matplotlib

To import TensorFlow and NSL, you can use the import statement as follows:

import tensorflow as tf # import TensorFlow library
import neural_structured_learning as nsl # import Neural Structured Learning library

To import NetworkX and Matplotlib, you can use the import statement as follows:

import networkx as nx # import NetworkX library
import matplotlib.pyplot as plt # import Matplotlib library

Now that you have installed and imported TensorFlow and other libraries, you are ready to load and preprocess the graph dataset. In the next section, you will learn how to load and preprocess the Cora dataset, which is a citation network of scientific papers.

3.2. Loading and preprocessing a graph dataset

To implement a graph neural network (GNN) with TensorFlow, you need a graph dataset that you can use to train and test your model. In this section, you will learn how to load and preprocess a graph dataset using the PyTorch Geometric (PyG) library, which is a popular and easy-to-use library for graph analysis with PyTorch.

PyG provides many built-in graph datasets that you can download and use for your experiments, such as the Cora, Citeseer, and Pubmed citation networks, the TU Datasets for graph classification, and the Open Graph Benchmark for various graph tasks. You can also load your own graph dataset from a file or a URL, as long as it follows the PyG data format.

In this tutorial, you will use the Cora citation network as an example. The Cora dataset consists of 2,708 scientific publications classified into one of seven classes. The citation network consists of 5,429 links. Each publication in the dataset is described by a 0/1-valued word vector indicating the absence/presence of the corresponding word from the dictionary. The dictionary consists of 1,433 unique words.

To load the Cora dataset, you need to import the torch_geometric.datasets module and use the Planetoid class, which is a generic class for handling the citation networks. You also need to specify the root directory where the dataset will be stored, and the name of the dataset. The following code snippet shows how to do this:

import torch_geometric.datasets as datasets

root = './data'
name = 'Cora'

dataset = datasets.Planetoid(root=root, name=name)

If the dataset is not already downloaded, PyG will automatically download it and save it in the root directory. The dataset object is a list of Data objects, where each Data object represents a single graph. In this case, the list contains only one graph, which is the whole citation network. You can access the graph by indexing the dataset object:

graph = dataset[0]

The graph object contains several attributes that describe the graph, such as x, edge_index, y, and train_mask. These attributes are explained below:

  • x: A tensor of shape [num_nodes, num_node_features] that stores the node feature vectors. In this case, x has shape [2708, 1433], which means that there are 2,708 nodes and 1,433 node features.
  • edge_index: A tensor of shape [2, num_edges] that stores the edge indices. Each column represents an edge, and the two rows represent the source and target nodes of the edge. In this case, edge_index has shape [2, 5429], which means that there are 5,429 edges.
  • y: A tensor of shape [num_nodes] that stores the node labels. In this case, y has shape [2708], which means that each node has a label from 0 to 6.
  • train_mask: A boolean tensor of shape [num_nodes] that stores the training mask. The training mask indicates which nodes are used for training the model. In this case, train_mask has shape [2708], and only 20 nodes per class are used for training.

Before you can use the graph dataset for training your model, you need to preprocess it to make it suitable for the GNN. One of the most common preprocessing steps is to add self-loops to the graph, which are edges that connect a node to itself. Self-loops can help the GNN to preserve the information of the nodes during the message passing process. To add self-loops to the graph, you need to import the torch_geometric.utils module and use the add_self_loops function, which takes the edge_index tensor as input and returns a new edge_index tensor with self-loops added. The following code snippet shows how to do this:

import torch_geometric.utils as utils

edge_index = graph.edge_index
edge_index_with_self_loops = utils.add_self_loops(edge_index)
graph.edge_index = edge_index_with_self_loops

Another common preprocessing step is to normalize the node features, which can help the GNN to converge faster and achieve better performance. To normalize the node features, you need to import the torch_geometric.transforms module and use the NormalizeFeatures class, which is a transform that normalizes the node feature matrix to have unit sum in each row. You can apply the transform to the graph by calling it as a function. The following code snippet shows how to do this:

import torch_geometric.transforms as transforms

normalize_features = transforms.NormalizeFeatures()
graph = normalize_features(graph)

Now you have a graph dataset that is ready to be used for training your GNN. In the next section, you will learn how to define and create a graph convolution layer, which is the main building block of your GNN.

3.3. Defining and creating a graph convolution layer

One of the main components of a graph neural network (GNN) is the graph convolution layer, which is a generalization of the standard convolution operation to graph-structured data. A graph convolution layer can aggregate the information from the nodes and their neighbors in a graph, and update the node feature vectors accordingly. In this section, you will learn how to define and create a graph convolution layer with TensorFlow using the Neural Structured Learning (NSL) framework.

NSL provides a GraphRegularization class, which is a wrapper that can add graph regularization to any Keras model. Graph regularization is a technique that leverages the graph structure of the data to improve the generalization and robustness of the model. NSL also provides a nsl.keras.layers.GraphConvolution class, which is a Keras layer that implements the graph convolution operation. You can use this class to create your own graph convolution layer, or use the predefined ones that NSL offers, such as nsl.keras.layers.GCN, nsl.keras.layers.GAT, or nsl.keras.layers.GraphSage.

In this tutorial, you will use the nsl.keras.layers.GCN class, which is a graph convolution layer based on the GCN model proposed by Kipf and Welling. A GCN layer can be defined as follows:

import neural_structured_learning as nsl

gcn_layer = nsl.keras.layers.GCN(units, activation, use_bias, kernel_regularizer, bias_regularizer, dropout_rate, name)

The parameters of the GCN layer are explained below:

  • units: An integer that specifies the number of output units of the layer. This determines the dimensionality of the updated node feature vectors.
  • activation: A string or a function that specifies the activation function of the layer. This determines how the output of the layer is transformed. You can use any activation function that is supported by Keras, such as 'relu', 'sigmoid', or 'softmax'.
  • use_bias: A boolean that specifies whether to use a bias term in the layer. This determines whether the layer has an additional parameter that is added to the output of the layer.
  • kernel_regularizer: A string or a function that specifies the regularizer for the kernel (or weight) matrix of the layer. This determines how the kernel matrix is penalized during the training process. You can use any regularizer that is supported by Keras, such as 'l1', 'l2', or 'l1_l2'.
  • bias_regularizer: A string or a function that specifies the regularizer for the bias vector of the layer. This determines how the bias vector is penalized during the training process. You can use any regularizer that is supported by Keras, such as 'l1', 'l2', or 'l1_l2'.
  • dropout_rate: A float between 0 and 1 that specifies the dropout rate of the layer. This determines how many units are randomly dropped out during the training process to prevent overfitting. A dropout rate of 0 means no dropout, and a dropout rate of 1 means complete dropout.
  • name: A string that specifies the name of the layer. This determines how the layer is identified in the model.

To create a GCN layer, you need to specify the values of these parameters according to your needs and preferences. For example, you can create a GCN layer with 16 output units, ReLU activation, bias term, L2 kernel regularization, no bias regularization, and 0.5 dropout rate, as follows:

gcn_layer = nsl.keras.layers.GCN(16, 'relu', True, 'l2', None, 0.5, 'gcn_layer')

Once you have created a GCN layer, you can use it to update the node feature vectors of your graph. To do this, you need to pass the node feature matrix and the normalized adjacency matrix of your graph to the GCN layer as inputs. The node feature matrix is a tensor of shape [num_nodes, num_node_features] that stores the node feature vectors. The normalized adjacency matrix is a tensor of shape [num_nodes, num_nodes] that stores the normalized edge weights of the graph. You can obtain the normalized adjacency matrix from the edge_index tensor by using the nsl.tools.normalize_graph function, which takes the edge_index tensor and the number of nodes as inputs and returns the normalized adjacency matrix. The following code snippet shows how to do this:

x = graph.x # node feature matrix of shape [num_nodes, num_node_features]
edge_index = graph.edge_index # edge index tensor of shape [2, num_edges]
num_nodes = x.shape[0] # number of nodes

# normalize the graph and get the normalized adjacency matrix
normalized_adjacency_matrix = nsl.tools.normalize_graph(edge_index, num_nodes)

# update the node feature vectors by passing them to the GCN layer
updated_x = gcn_layer([x, normalized_adjacency_matrix]) # updated node feature matrix of shape [num_nodes, units]

Now you have learned how to define and create a graph convolution layer with TensorFlow and NSL. In the next section, you will learn how to build and train a graph neural network model using the graph convolution layer and the graph dataset that you have prepared.

3.4. Building and training a graph neural network model

Now that you have learned how to define and create a graph convolution layer with TensorFlow and Neural Structured Learning (NSL), you are ready to build and train a graph neural network (GNN) model using the graph convolution layer and the graph dataset that you have prepared. In this section, you will learn how to do this step by step.

The first step is to define the architecture of your GNN model. A GNN model consists of one or more graph convolution layers, followed by an optional global pooling layer and a final output layer. The graph convolution layers are responsible for updating the node feature vectors based on the graph structure and the node features. The global pooling layer is responsible for aggregating the node feature vectors into a single graph feature vector, which can be used for graph-level tasks. The output layer is responsible for producing the desired output of the model, such as node labels, edge weights, or graph labels.

In this tutorial, you will build a simple GNN model for the node classification task, which aims to assign a label to each node in a graph based on its features and neighborhood. Your GNN model will have two graph convolution layers, each with 16 output units and ReLU activation, followed by a softmax output layer that produces a probability distribution over the node labels. You will use the nsl.keras.layers.GCN class to create the graph convolution layers, and the tf.keras.layers.Dense class to create the output layer. You will also use the tf.keras.Model class to define the model as a subclass, and override the call method to specify the forward pass of the model. The following code snippet shows how to do this:

import tensorflow as tf
import neural_structured_learning as nsl

class GNNModel(tf.keras.Model):
  def __init__(self, num_classes, name='gnn_model'):
    super(GNNModel, self).__init__(name=name)
    # create the graph convolution layers
    self.gcn1 = nsl.keras.layers.GCN(16, 'relu', True, 'l2', None, 0.5, 'gcn1')
    self.gcn2 = nsl.keras.layers.GCN(16, 'relu', True, 'l2', None, 0.5, 'gcn2')
    # create the output layer
    self.output_layer = tf.keras.layers.Dense(num_classes, 'softmax', name='output_layer')

  def call(self, inputs):
    # unpack the inputs
    x, normalized_adjacency_matrix = inputs
    # pass the inputs through the graph convolution layers
    x = self.gcn1([x, normalized_adjacency_matrix])
    x = self.gcn2([x, normalized_adjacency_matrix])
    # pass the output of the last graph convolution layer to the output layer
    output = self.output_layer(x)
    return output

The second step is to create an instance of your GNN model and compile it. To create an instance of your GNN model, you need to specify the number of classes that you want to predict. In this case, you want to predict the node labels in the Cora dataset, which have seven possible values. Therefore, you need to set the number of classes to seven. To compile your GNN model, you need to specify the optimizer, the loss function, and the metrics that you want to use to train and evaluate your model. In this case, you will use the Adam optimizer, the sparse categorical cross-entropy loss, and the accuracy metric. The following code snippet shows how to do this:

# create an instance of the GNN model
gnn_model = GNNModel(7)

# compile the GNN model
gnn_model.compile(optimizer='adam', loss='sparse_categorical_crossentropy', metrics=['accuracy'])

The third step is to train your GNN model on the graph dataset. To train your GNN model, you need to pass the node feature matrix, the normalized adjacency matrix, and the node labels to the fit method of the model. You also need to specify the number of epochs, the batch size, and the validation data. In this case, you will train your GNN model for 200 epochs, with a batch size of 2708 (which is the number of nodes in the graph), and using the validation mask to split the data into training and validation sets. The validation mask is a boolean tensor of shape [num_nodes] that indicates which nodes are used for validation. The following code snippet shows how to do this:

# get the node feature matrix, the normalized adjacency matrix, and the node labels from the graph
x = graph.x
normalized_adjacency_matrix = nsl.tools.normalize_graph(graph.edge_index, x.shape[0])
y = graph.y

# get the training mask and the validation mask from the graph
train_mask = graph.train_mask
val_mask = graph.val_mask

# train the GNN model on the graph dataset
gnn_model.fit([x, normalized_adjacency_matrix], y, epochs=200, batch_size=2708, validation_data=([x, normalized_adjacency_matrix], y, val_mask), sample_weight=train_mask)

The fourth and final step is to evaluate your GNN model on the graph dataset. To evaluate your GNN model, you need to pass the node feature matrix, the normalized adjacency matrix, and the node labels to the evaluate method of the model. You also need to specify the batch size and the test data. In this case, you will evaluate your GNN model with a batch size of 2708, and using the test mask to split the data into training and test sets. The test mask is a boolean tensor of shape [num_nodes] that indicates which nodes are used for testing. The following code snippet shows how to do this:

# get the test mask from the graph
test_mask = graph.test_mask

# evaluate the GNN model on the graph dataset
gnn_model.evaluate([x, normalized_adjacency_matrix], y, batch_size=2708, sample_weight=test_mask)

Congratulations! You have successfully built and trained a graph neural network model with TensorFlow and NSL. In the next section, you will learn how to evaluate and visualize the model performance on the graph dataset.

3.5. Evaluating and visualizing the model performance

After you have trained your graph neural network (GNN) model with TensorFlow and Neural Structured Learning (NSL), you may want to evaluate and visualize its performance on the graph dataset. In this section, you will learn how to do this using some useful tools and libraries.

The first tool that you will use is the tf.keras.Model.evaluate method, which you have already used in the previous section to evaluate your GNN model on the test set. This method returns the loss and the metrics that you have specified when you compiled your model. In this case, you have used the sparse categorical cross-entropy loss and the accuracy metric. The following code snippet shows how to print the loss and the accuracy of your GNN model on the test set:

# get the test mask from the graph
test_mask = graph.test_mask

# evaluate the GNN model on the test set
loss, accuracy = gnn_model.evaluate([x, normalized_adjacency_matrix], y, batch_size=2708, sample_weight=test_mask)

# print the loss and the accuracy
print('Test loss:', loss)
print('Test accuracy:', accuracy)

The second tool that you will use is the tf.keras.Model.predict method, which returns the predictions of your GNN model for the given inputs. This method can be useful if you want to analyze the output of your model in more detail, such as the probability distribution over the node labels, or the predicted label for each node. The following code snippet shows how to get the predictions of your GNN model for the whole graph, and how to print the predicted label for a specific node:

# get the predictions of the GNN model for the whole graph
predictions = gnn_model.predict([x, normalized_adjacency_matrix])

# print the shape of the predictions
print('Predictions shape:', predictions.shape)

# print the predictions for the first node
print('Predictions for the first node:', predictions[0])

# get the predicted label for the first node by taking the argmax of the predictions
predicted_label = tf.argmax(predictions[0])

# print the predicted label for the first node
print('Predicted label for the first node:', predicted_label)

The third tool that you will use is the Matplotlib library, which is a popular and powerful library for creating and displaying various types of plots and figures. You can use Matplotlib to visualize the performance of your GNN model in different ways, such as the loss and the accuracy curves, the confusion matrix, or the node embeddings. The following code snippet shows how to import Matplotlib and how to plot the loss and the accuracy curves of your GNN model during the training process:

# import Matplotlib
import matplotlib.pyplot as plt

# get the history object from the GNN model
history = gnn_model.history

# get the loss and the accuracy values from the history object
loss_values = history.history['loss']
val_loss_values = history.history['val_loss']
accuracy_values = history.history['accuracy']
val_accuracy_values = history.history['val_accuracy']

# get the number of epochs from the history object
epochs = range(1, len(history.epoch) + 1)

# plot the loss curve
plt.plot(epochs, loss_values, 'bo', label='Training loss')
plt.plot(epochs, val_loss_values, 'b', label='Validation loss')
plt.title('Training and validation loss')
plt.xlabel('Epochs')
plt.ylabel('Loss')
plt.legend()
plt.show()

# plot the accuracy curve
plt.plot(epochs, accuracy_values, 'ro', label='Training accuracy')
plt.plot(epochs, val_accuracy_values, 'r', label='Validation accuracy')
plt.title('Training and validation accuracy')
plt.xlabel('Epochs')
plt.ylabel('Accuracy')
plt.legend()
plt.show()

These are some of the tools and libraries that you can use to evaluate and visualize the performance of your GNN model on the graph dataset. You can also use other tools and libraries that suit your needs and preferences, such as Seaborn, Plotly, or Scikit-learn. The main goal is to understand how well your model performs on the graph data, and how you can improve it further.

You have reached the end of this section, and the end of this tutorial. In the next and final section, you will summarize what you have learned in this tutorial, and provide some suggestions for further learning and exploration.

4. Conclusion and Future Work

In this tutorial, you have learned how to implement a graph neural network (GNN) with TensorFlow and Neural Structured Learning (NSL) and apply it to a graph analysis problem. You have covered the following topics:

  • What are graphs and graph neural networks, and why they are useful for graph analysis.
  • How to install and import TensorFlow and other libraries that you need for this tutorial.
  • How to load and preprocess a graph dataset using the PyTorch Geometric library.
  • How to define and create a graph convolution layer using the NSL framework.
  • How to build and train a GNN model using the graph convolution layer and the graph dataset.
  • How to evaluate and visualize the performance of your GNN model using various tools and libraries.

By following this tutorial, you have gained a basic understanding of the concepts and techniques of graph neural networks, and how to use them in practice with TensorFlow and NSL. You have also seen an example of how to solve a node classification task on the Cora citation network using a GNN model.

However, this tutorial is only an introduction, and there is much more to learn and explore about graph neural networks and graph analysis. Here are some suggestions for further learning and exploration:

  • Try different graph datasets and tasks, such as link prediction, graph clustering, graph generation, or graph representation learning. You can find many graph datasets and tasks on the Open Graph Benchmark website.
  • Try different graph neural network architectures, such as graph attention networks, graph sage networks, or graph isomorphism networks. You can find many graph neural network architectures on the PyTorch Geometric website.
  • Try different hyperparameters and regularization techniques, such as the number of layers, the number of units, the activation function, the dropout rate, the learning rate, or the graph regularization coefficient. You can use the TensorBoard tool to monitor and compare the results of different experiments.
  • Read more about the theory and applications of graph neural networks and graph analysis. You can find many resources on the GNNPapers GitHub repository.

We hope that you have enjoyed this tutorial, and that you have learned something new and useful. Thank you for reading, and happy graphing!

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