Overview of DANMF (Dynamic Attributed Network with Matrix Factorization) and examples of implementation

DANMF (Dynamic Attributed Network with Matrix Factorization)

DANMF (Dynamic Attributed Network with Matrix Factorization) is one of the graph embedding methods for network data with dynamic attribute information of nodes and network structure. The method learns to embed nodes by combining node attribute information with the network structure. This method is particularly useful when dynamic attribute information is included, and is suitable when node attributes change with time or when different attribute information is available at different time steps. The main features and elements of DANMF are described below.

1. dynamic attribute information:

DANMF is designed to handle cases where node attribute information changes over time, and if a node’s attributes vary at different time steps, DANMF takes this into account when learning embeddings.

2. graph structural information:

DANMF also utilizes network structure information to learn graph embedding, taking into account the connection relationships among nodes and the weights of edges.

3. Matrix Factorization:

DANMF learns embeddings using matrix factorization. Specifically, DANMF decomposes the node attribute matrix and network structure matrix into low-dimensional matrices and combines these matrices to generate embeddings.

4. learning at different time steps:

DANMF learns embeddings at each time step in order to account for data at different time steps. This makes it suitable for cases where node attribute information varies with time.

5. introducing non-linearity:

DANMF can also learn embeddings using nonlinear functions, which allows it to acquire a richer representation.

As a graph embedding method for network data with dynamic attribute information, DANMF is useful in applications such as time series data and social networks, and can effectively learn node embeddings by taking into account changes in attribute information over time and the effects of network structure. It is a method.

Specific procedures for DANMF

The basic procedures of DANMF are described below.

1. Data Preparation:

Collect or prepare graph data and node attribute information. Graph data should consist of nodes and edges, and node attribute information should be available at each time step.

2. matrix construction:

Convert node attribute information and network structure information into matrix form. Typically, a node attribute matrix (number of dimensions of attributes x number of nodes) and a network structure matrix (number of nodes x number of nodes) are created, with the attribute matrix having a different form for each time step.

3. matrix factoring:

Matrix factorization is used to decompose the attribute and network structure matrices into low-dimensional matrices. Typically, matrix factorization is performed to perform dimensionality reduction, yielding an embedded matrix (a low-dimensional representation). The decomposition uses methods such as Singular Value Decomposition (SVD) as described in “Overview of Singular Value Decomposition (SVD) and examples of algorithms and implementations“.

4. learning the embedding at each time step:

At each time step, the decomposed embedding matrix is learned. At each time step, the attribute matrix and the network structure matrix are different, and the embedding matrix is learned by considering both attribute and network structure information.

5. embedding merging:

The embedding matrices learned at each time step are combined to obtain a consistent embedding representation across time steps. The general approach is to combine embedding matrices to form a new embedding matrix.

6. task-specific use:

The learned embeddings are applied to specific tasks such as class classification, link prediction, clustering, and recommendation. The resulting embeddings will help to capture similarities between nodes.

7. hyper-parameter tuning:

Tune hyperparameters such as the number of dimensions and learning rate of matrix factorization and perform trial and error to obtain optimal model performance.

Example Implementation of DANMF

import numpy as np
from numpy.linalg import inv

# Dummy graph data (network structure matrix)
# In this example, we consider three nodes and two time steps
A1 = np.array([[0, 1, 0],
               [1, 0, 1],
               [0, 1, 0]])

A2 = np.array([[0, 1, 1],
               [1, 0, 0],
               [1, 0, 0]])

# Dummy attribute data (attribute matrix)
# Consider different attribute information at each time step
X1 = np.array([[0.1, 0.2],
               [0.3, 0.4],
               [0.5, 0.6]])

X2 = np.array([[0.2, 0.3],
               [0.4, 0.5],
               [0.6, 0.7]])

# hyperparameter
embedding_dim = 2
lambda_reg = 0.01  # Weights of regularization terms

# Initialization of embedding matrix
H1 = np.random.rand(X1.shape[0], embedding_dim)
H2 = np.random.rand(X2.shape[0], embedding_dim)

# DANMF study (at each time step)
for i in range(100):  # Number of iterations
    # Update time step 1
    H1 = np.dot(np.dot(X1.T, X1) + lambda_reg * np.eye(embedding_dim),
                np.dot(X1.T, A1).T)
    
    # Update time step 2
    H2 = np.dot(np.dot(X2.T, X2) + lambda_reg * np.eye(embedding_dim),
                np.dot(X2.T, A2).T)

# Combining embedded matrices (per time step)
H_combined = np.concatenate((H1, H2), axis=0)

# Perform tasks using the resulting embedding (class classification, link prediction, etc.)

# Here is the code to add the task