Overview of relational data learning and examples of applications and implementations

Overview of Relational Data Learning

Relational Data Learning (RDA) is a machine learning method for relational data (e.g., graphs, networks, tabular data, etc.). While traditional machine learning is usually applied only to individual instances (e.g., vectors or matrices), relational data learning considers multiple instances and the relationships among them.

Relational data learning treats nodes and edges of a graph or network, or tables and their relationships in a database, as the basic structure of the data, thereby enabling more complex patterns and features to be extracted and data to be analyzed and predicted using the relationships and context among the instances This will allow for more complex patterns and features to be extracted and data to be analyzed and predicted.

Application examples of relational data learning

Relational data learning has been widely applied in a variety of domains. Below are some specific applications of relational data learning.

• Social network analysis: Social networks contain graph data representing relationships among users. By using relational data learning, it is possible to perform friend recommendation, influence prediction, and modeling of information diffusion based on user attributes and relationships.
• Recommendation system: A recommendation system models the relationship between users and items. By applying relational data learning to this model, personalized recommendations can be made based on user preferences and item characteristics.
• Bioinformatics: In bioinformatics, relational data such as protein and gene interaction networks and metabolic pathways play an important role. By using relational data learning, it is possible to predict protein functions, identify disease-related genes, and search for new drug targets.
• Natural Language Processing: In natural language processing, relational data learning is used to model the relationships among words, phrases, and sentences in a sentence. This enables modeling that takes into account sentence structure and semantic relationships in tasks such as document classification, machine translation, and information extraction.
• Database management: Relational databases model the relationships between different tables. By applying relational data learning to this, tasks such as database query optimization and data integration become possible.
• Information Retrieval: Information retrieval can be made more efficient by extracting features from the attributes in relational data and converting them into an appropriate form of expression to make it easier to express the meaning and relationships of the data. It can also learn patterns and relationships in relational data to retrieve data that match the conditions given as a query, predict relevant data, and optimize query execution plans by using statistical information about the tables and queries in the database.

Next, we describe the algorithms used for relational data learning.

Algorithms used for relational data learning

Various algorithms and methods are used in relational data learning, including

• Spectral clustering: In spectral clustering, data points are represented as vertices in a graph, similarity between data points is represented as edge weights, the Laplacian matrix of the graph is calculated, the eigenvectors are used to embed the data in a low-dimensional space, and the embedded data are The method will cluster the data using a clustering algorithm (e.g., k-means).
• Matrix Factorization: Matrix factorization is a method for decomposing a matrix into a product of several submatrices. This method is used for data dimensionality reduction, data completion, feature extraction, etc. Typical methods include Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) Non-negative Matrix Factorization (NMF), etc.
• Tensor Decomposition: Tensor decomposition is a method for decomposing relational data into low-rank tensor approximations. Tensor decomposition is widely used in recommendation systems and co-occurrence data analysis. Typical methods include CP decomposition, Tucker decomposition described in “Tucker Decomposition Overview, Algorithm, and Implementation Examples“, and HOSVD.
• Stochastic Block Model (SBM): In SBM, a set of nodes is partitioned into several blocks (groups), and a graph is generated by defining the joining probabilities between blocks and the generation probabilities of edges.
• Graph Neural Networks: Graph neural networks are learning methods for graph data. This method updates node and edge information through bi-directional information propagation and extracts features of the entire graph. Typical models include Graph Convolutional Networks (GCN), GraphSAGE described in “GraphSAGE Overview, Algorithm, and Example Implementation“, and GIN.
• Graph Convolutional Networks: Graph Convolutional Networks apply convolutional neural networks to graph data. This method extracts features by considering adjacent nodes and their relationships in a graph, and is used for tasks such as node and graph classification, prediction, and clustering.
• Graph Embedding: Graph embedding is a technique that embeds graph data into a low-dimensional vector space. This allows features of nodes and edges of the graph to be treated as vector representations. Typical methods include Node2Vec described in “Overview of Node2Vec, its algorithm and implementation examples“, DeepWalk described in “DeepWalk Overview, Algorithms, and Example Implementations,”, and GraphSAGE described in “Overview of GraphSAGE and Examples of Algorithms and Implementations“.
• Meta-Path Walk: Meta-Path Walk is a method for propagating information on specific paths in graph data. This method selects specific metapaths (patterns of connections between different nodes) and extracts node features based on them, and has been applied to tasks such as recommendation systems and information retrieval.

The following sections describe the details of these algorithms and examples of their implementations.

Spectral Clustering

Spectral clustering can be a very effective method for clustering data sets. In this technique, clustering is performed using a graph structure that represents the proximity and similarity of data. In spectral clustering, data points are represented as vertices of a graph, similarity between data points is represented as edge weights, the Laplacian matrix of the graph is computed, and the eigenvectors are used to embed the data in a low-dimensional space. Finally, the embedded data shall be clustered using a clustering algorithm (e.g., k-means).

The procedure for spectral clustering is as follows

1. Create a distance matrix (or similarity matrix) to calculate the similarity between data points. Common distance measures include Euclidean distance and cosine similarity.
2. The distance matrix is used to construct a graph. The graph has each data point as a vertex and the elements of the distance matrix as edge weights. There are k nearest neighbor and epsilon-nearest neighbor methods for graph construction.
3. Calculate the Laplacian matrix of the graph. The Laplacian matrix is the difference between the adjacency matrix and the order matrix of the graph.
4. Compute the eigenvectors of the Laplacian matrix. In general, the eigenvectors corresponding to the smallest eigenvalues provide information about the connected components of the graph.
5. The computed eigenvectors are embedded in a low-dimensional space. This allows low-dimensional features corresponding to the major eigenvectors to be obtained.
6. The obtained low-dimensional features are clustered using a clustering algorithm (e.g. k-means).

Spectral clustering can be implemented using the Python and scikit-learn libraries. An example of spectral clustering using scikit-learn is shown below.

from sklearn.cluster import SpectralClustering
from sklearn.datasets import make_blobs

# Creating a dummy data set
X, _ = make_blobs(n_samples=200, centers=3, random_state=0)

# Perform spectral clustering
clustering = SpectralClustering(n_clusters=3, assign_labels="discretize", random_state=0).fit(X)

# Display clustering results
print(clustering.labels_)

In the above code, the make_blobs function is used to generate a dummy data set with three clusters, and the SpectralClustering class is used to perform spectral clustering. n_clusters parameter specifies the number of clusters, and assign _labels parameter specifies how to assign labels to the clustering results.

Spectral clustering has been applied to a variety of tasks such as image segmentation and clustering of graph data because of its ability to capture the nonlinear structure of data.

Matrix Factorization Method

Matrix Factorization is a method of decomposing a matrix into a product of several submatrices. This method is used for data dimensionality reduction, data completion, feature extraction, etc. Typical matrix factorization methods include Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF).

Singular Value Decomposition (SVD) decomposes a real or complex-valued matrix into a product of three matrices, and for a given matrix A, the decomposition is as follows

A = U * Σ * V^T

where U and V are orthogonal matrices, Σ is a diagonal matrix, and the diagonal elements are called singular values. Singular value decomposition is sometimes used for data dimensionality reduction or data completion.

Non-negative matrix factorization (NMF) decomposes a non-negative matrix into the form of a product of non-negative submatrices, which, for a given non-negative matrix A, shall be decomposed as follows.

A ≈ WH

where W and H are non-negative matrices and can be considered as approximate representations of A. NMF may be applied to tasks such as topic modeling and feature extraction.

For these specific implementations, matrix factorization can be implemented in Python using a numerical library such as NumPy or SciPy. Below are examples of implementing singular value decomposition (SVD) and nonnegative matrix factorization (NMF) using NumPy.

Example implementation of Singular Value Decomposition (SVD):

import numpy as np

# original matrix
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# singular value analysis
U, S, VT = np.linalg.svd(A)

# decomposed matrix
print("U:", U)
print("S:", S)
print("VT:", VT)

Example implementation of non-negative matrix factorization (NMF):

import numpy as np
from sklearn.decomposition import NMF

# original matrix
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# non-negative matrix factorization
model = NMF(n_components=2)
W = model.fit_transform(A)
H = model.components_

# decomposed matrix
print("W:", W)
print("H:", H)

The above code uses NumPy to perform singular value decomposition (SVD) and nonnegative matrix factorization (NMF). In the singular value decomposition, the matrix is decomposed using the np.linalg.svd function, and in the nonnegative matrix factorization, the matrix is decomposed using the sklearn.decomposition.NMF class. At the end of each, the resulting decomposed matrix is displayed.

Tensor Factorization

Tensor Factorization (TF) is a method for decomposing a high-dimensional tensor into the form of a product of several low-rank tensors. A tensor is a generalization of a multidimensional array, and a matrix is a special case of a two-dimensional tensor. Tensor decomposition is used for data dimensionality reduction, feature extraction, and data completion. Typical tensor decomposition methods include the CANDECOMP/PARAFAC (CP) decomposition and the non-negative value tensor decomposition (NMF).

The CANDECOMP/PARAFAC(CP) decomposition decomposes a given tensor into the form of a product of multiple tensors, which for a given tensor A is decomposed as follows.

A = [A^(1), A^(2), … , A^(n)]

where A^(n) is the nth tensor and N is the number of dimensions of the tensor CP decomposition may be used for tensor feature extraction and data completion.

Non-negative tensor decomposition (NMF) decomposes a non-negative tensor into a product form of non-negative low-rank tensors, which, for a given non-negative tensor A, shall be decomposed as follows.

A ≈ [W^(1), W^(2), . , W^(N)]

where W^(n) is a non-negative, low-rank tensor and can be considered as an approximate representation of A. NMF is applied to tasks such as feature extraction and topic modeling of multidimensional data.

Tensor decomposition is useful in high-dimensional data analysis, and tensor libraries and machine learning libraries provide implementations of tensor decomposition; in Python, deep learning frameworks such as TensorFlow and PyTorch, as well as Tensorly and TensorLab Tensor decomposition can be implemented using tensor-specific libraries.

An example implementation of tensor decomposition using the Tensorly library is shown below.

import tensorly as tl
from tensorly.decomposition import parafac

# Generating Tensors
X = tl.tensor([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])

# Tensor decomposition by CP decomposition
factors = parafac(X, rank=2)

# Element display of the decomposed tensor
for factor in factors:
    print(factor)

The above code uses the Tensorly library to perform tensor decomposition, using the tl.tensor function to generate the input tensor X and the parafac function to perform CP decomposition. the rank (number of dimensions) after decomposition is specified by the rank parameter. The tensor elements after decomposition are stored in factors, each of which represents a component of the decomposed tensor.

Tensorly is a library that allows easy manipulation and decomposition of tensors, but other libraries (e.g. PyTorch and TensorFlow) also provide similar functions.

Stochastic Block Model;SBM

The Stochastic Block Model (SBM) is a kind of generative model for graph data, in which a set of nodes is partitioned into several blocks (groups) and a graph is generated by defining the probabilities of joining the blocks and generating edges. SBM has been applied to problems such as network analysis, graph clustering, and community detection.

In SBM, the following elements must be specified

• Number of nodes: Specify the total number of nodes in the graph.
• Number of blocks: Specify the number of blocks (groups) into which the graph is divided.
• Joint probability between blocks: A matrix that represents the probability of generating edges between each block. The elements of the matrix indicate the probability of generating edges between blocks.
• Joint probability within a block: A vector representing the probability of generating an edge within each block is specified. The elements of the vector indicate the probability of generating edges between nodes in the block.

Python has a variety of libraries that can be used to generate graphs and implement SBM, or you can use popular graph libraries such as NetworkX or igraph.

Below is an example implementation of SBM using NetworkX.

import networkx as nx
import numpy as np

# Specify the number of blocks and the number of nodes
num_blocks = 3
num_nodes = 100

# Specify join probability between blocks
p_matrix = np.array([[0.8, 0.2, 0.1],
                    [0.2, 0.7, 0.3],
                    [0.1, 0.3, 0.9]])

# Specify the join probability within a block
p_vector = np.array([0.5, 0.4, 0.3])

# Generate graphs based on SBM
block_sizes = [num_nodes // num_blocks] * num_blocks
graph = nx.stochastic_block_model(block_sizes, p_matrix, p_vector)

# Graph Visualization
nx.draw(graph, with_labels=True)

In the above code, the NetworkX library is used to generate a graph based on SBM: num_blocks specifies the number of blocks, num_nodes the total number of nodes, p_matrix the joint probability between blocks, and p_vector the joint probability within blocks. The nx.stochastic_block_model function is used to generate a graph based on SBM, and the nx.draw function is used to visualize the graph.

Graph Neural Network;GNN

Graph Neural Networks (GNNs) are neural networks that take a graph structure as input and learn node and edge features. classification, etc.).

GNNs use information from neighboring nodes and edges to represent node and edge features, and a typical GNN architecture consists of a step of aggregating node neighbor information and a step of updating the aggregated information. This process is repeated multiple times to learn a feature representation of the entire graph.

Python has a variety of libraries that can be used to implement GNNs. Below is an example of a GNN implementation using PyTorch Geometric.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.nn import MessagePassing
from torch_geometric.utils import add_self_loops, degree

class GraphConvolution(MessagePassing):
    def __init__(self, in_channels, out_channels):
        super(GraphConvolution, self).__init__(aggr='add')
        self.linear = nn.Linear(in_channels, out_channels)

    def forward(self, x, edge_index):
        # Add self-loops to graph edges
        edge_index, _ = add_self_loops(edge_index, num_nodes=x.size(0))
        # Calculate the degree of a node
        row, col = edge_index
        deg = degree(row, x.size(0), dtype=x.dtype)
        # Computing Graph Convolution
        x = self.linear(x)
        x = self.propagate(edge_index, x=x, deg=deg)
        return x

    def message(self, x_j, deg):
        return x_j / deg.unsqueeze(-1)

class GNN(nn.Module):
    def __init__(self, in_channels, hidden_channels, out_channels):
        super(GNN, self).__init__()
        self.conv1 = GraphConvolution(in_channels, hidden_channels)
        self.conv2 = GraphConvolution(hidden_channels, out_channels)

    def forward(self, x, edge_index):
        x = F.relu(self.conv1(x, edge_index))
        x = self.conv2(x, edge_index)
        return x

In the above code, GNN is implemented using the PyTorch Geometric library: the GraphConvolution class represents a graph convolution layer for updating node features, and the GNN class represents a GNN model combining multiple graph convolution layers. When training a GNN model, the node feature matrix x and edge index edge_index are given as input, and forward propagation is performed using the forward method.

Other GNN libraries include Deep Graph Library (DGL), Spektral, and StellarGraph.

Graph Convolutional Network;GCN

A Graph Convolutional Network (GCN) is a neural network for performing convolutional operations on graph data. GCNs are widely used to solve tasks on graphs by updating node features using node and edge information. It is widely used to solve tasks on graphs.

In a GCN, the feature representation of a node is updated using information from neighboring nodes. Specifically, the feature representation of each node is computed via operations such as weighted averaging and concatenation of the feature representations of neighboring nodes. This exchange of information with neighboring nodes allows the feature representation of a node to reflect information about a more global graph structure.

A simple example implementation of GCN using PyTorch is shown below.

import torch
import torch.nn as nn
import torch.nn.functional as F

class GraphConvolution(nn.Module):
    def __init__(self, in_features, out_features):
        super(GraphConvolution, self).__init__()
        self.linear = nn.Linear(in_features, out_features)

    def forward(self, x, adj):
        x = torch.matmul(adj, self.linear(x))
        x = F.relu(x)
        return x

class GCN(nn.Module):
    def __init__(self, in_features, hidden_features, out_features):
        super(GCN, self).__init__()
        self.conv1 = GraphConvolution(in_features, hidden_features)
        self.conv2 = GraphConvolution(hidden_features, out_features)

    def forward(self, x, adj):
        x = self.conv1(x, adj)
        x = self.conv2(x, adj)
        return x

In the above code, the GraphConvolution class represents the graph convolution layer, and the GCN class defines a GCN model that combines two graph convolution layers. When training the model, the node feature matrix x and the adjacency matrix adj are given as inputs, and forward propagation is performed using the forward method. in GraphConvolution, the feature representation of a node is updated by taking the matrix product of the adjacency matrix and the node feature matrix.

Note that libraries such as PyTorch Geometric and DGL also provide implementations of GCN, and more advanced features and efficient operations are available.

Graph Embedding

Graph Embedding is a technology that transforms graph data into a low-dimensional vector space. By converting graph structure and node information into a compact vector representation, graph embedding can extract features useful for graph data analysis and machine learning tasks.

There are a variety of graph embedding methods, and some representative methods are described below.

• Random Walk-based Methods: Random walk described in “Overview of Random Walks, Algorithms, and Examples of Implementations” on the graph are used to collect information about the neighborhood of a node and learn node embeddings based on that information. Typical methods include DeepWalk and Node2Vec.
• Spectral Methods: These methods use eigenvalues and eigenvectors of the Laplacian matrix and adjacency matrix of a graph to learn embeddings that capture the properties of the graph. Typical methods include Graph Laplacian and GraphSAGE.
• Graph Convolutional Neural Networks (GCNs): GCNs learn to embed nodes by performing convolutional operations on graph data. features based on the graph structure can be effectively extracted. see “Overview, Algorithms, and Application of Graph Convolutional Neural Networks (GCNs)” for details
• Graph Autoencoders: These learn to embed graphs by taking graphs as input and reconstructing the graph data using an Autoencoder. This allows the acquisition of low-dimensional vector representations while preserving the graph structure and node characteristics.

Among these methods, the method to be applied should be selected according to the nature of the graph data and the applied task. in Python, many libraries provide implementations of graph embedding. typical libraries include NetworkX, DeepWalk, Node2Vec, PyTorch Geometric, and DGL.

Below we describe some libraries and example implementations of graph embedding methods using Python.

NetworkX: a library for graph embedding in Python

import networkx as nx
import numpy as np
from gensim.models import Word2Vec

# Loading Graph Data
G = nx.read_edgelist('graph_data.txt')

# Graph Embedding Learning with DeepWalk
walks = nx.simulate_walks(G, num_walks=10, walk_length=80)
walks = [list(map(str, walk)) for walk in walks]
model = Word2Vec(walks, size=128, window=5, min_count=0, sg=1, workers=4)

# Obtaining the embedding vector of a node
node_embeddings = {node: model.wv[str(node)] for node in G.nodes()}

PyTorch Geometric:

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.nn import GCNConv
from torch_geometric.datasets import Planetoid
from torch_geometric.utils import to_networkx

# Loading Data Sets
dataset = Planetoid(root='data', name='Cora')
G = to_networkx(dataset[0])

# Graph Embedding Learning with GCN
class GCN(nn.Module):
    def __init__(self, in_features, out_features):
        super(GCN, self).__init__()
        self.conv1 = GCNConv(in_features, 16)
        self.conv2 = GCNConv(16, out_features)

    def forward(self, x, edge_index):
        x = F.relu(self.conv1(x, edge_index))
        x = self.conv2(x, edge_index)
        return x

model = GCN(dataset.num_features, 128)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
criterion = nn.CrossEntropyLoss()

def train():
    model.train()
    optimizer.zero_grad()
    output = model(dataset.x, dataset.edge_index)
    loss = criterion(output[dataset.train_mask], dataset.y[dataset.train_mask])
    loss.backward()
    optimizer.step()

# learning loop
for epoch in range(200):
    train()

# Obtaining the embedding vector of a node
with torch.no_grad():
    model.eval()
    node_embeddings = model(dataset.x, dataset.edge_index).detach().numpy()

Meta-path Walk

Meta-path Walk (Meta-path Walk) will be a method used to represent patterns and relationships between different nodes in graph data. In a metapath walk, a sequence called a metapath is defined to represent a specific pattern, and a random walk is performed on the graph based on the metapath. This allows patterns between different nodes to be collected and a feature representation of the graph data to be created.

Below is an example implementation of a metapath walk using Python.

import networkx as nx
import random

def meta_path_walk(graph, meta_path, start_node, walk_length):
    walk = [start_node]

    # Execution of random walk
    for _ in range(walk_length - 1):
        current_node = walk[-1]
        neighbors = graph.neighbors(current_node)

        # Select next node based on metapath
        next_node = random.choice([n for n in neighbors if graph.has_edge(n, current_node, key=meta_path[-1])])

        walk.append(next_node)

    return walk

# Loading Graph Data
G = nx.read_edgelist('graph_data.txt')

# Metapath Definition
meta_path = ['A', 'B', 'A']

# Execute metapath walk
start_node = 'node1'
walk_length = 10
walk = meta_path_walk(G, meta_path, start_node, walk_length)

In the above code, the NetworkX library is used to read graph data, and the meta_path_walk function executes a random walk according to the specified metapath and returns a walk of the specified length. The specific metapath must be defined according to the graph data used and the task to be analyzed. The metapath is designed to reflect a specific pattern on the graph by specifying the types of nodes and edges.

Metapath walks are used to extract relationships and patterns among different nodes in graph data. This technique is applied to graph-based reasoning and feature extraction and is used in a variety of applications, for example, in recommendation systems and social network analysis.

Reference Information and Reference Books

Detailed information on relational data learning is provided in “Relational Data Learning“. Please refer to it as well.

Reference books include “Relational Data Mining”

“Inference and Learning Systems for Uncertain Relational Data“

“Graph Neural Networks: Foundations, Frontiers, and Applications“

“Hands-On Graph Neural Networks Using Python: Practical techniques and architectures for building powerful graph and deep learning apps with PyTorch“

“Matrix Algebra“

“Non-negative Matrix Factorization Techniques: Advances in Theory and Applications“

“An Improved Approach On Distortion Decomposition Of Magnetotelluric Impedance Tensor“