Overview of embedding of dynamic graphs, algorithms and implementation examples

Overview of Dynamic Graph Embedding

Graph convolution described in “Overview, Algorithms, and Examples of Graph Convolutional Neural Networks (GCN)“, graph embedding described in “Overview, Algorithms, and Examples of Graph Embedding (GCN)“, and graph autoencoders described in “Overview, Algorithms, and Examples of Graph Deep Network Embedding (SDNE)” are mainly used to build models for static graphs. In contrast, there are many relationships represented by dynamic graphs in real data, and various approaches have been taken as described in “Methods for Analyzing Graph Data that Changes with Time.

To embed a dynamic graph, it is necessary to create a vector representation that reflects both the structure of the graph and its dynamic changes.

• Discrete-time dynamic graph (DTDG): A sequence of graphs that represent snapshots of dynamic changes in a graph at regular time intervals (above left).
• Continuous-time dynamic graph (CTDG): A graph in which each node or edge is time-stamped with its appearance and disappearance times (above right).

The difference between conventional embedding for static graphs and dynamic graph embedding is that conventional embedding for static graphs focuses on obtaining a fixed representation of nodes, while embedding for dynamic graphs aims to obtain a representation that corresponds to temporal changes in the graph. point.

There are several approaches to dynamic graph embedding

Snapshot-based methods:

1. Static Embedding + Temporal Alignment: This method uses a static graph embedding technique to obtain the embedding for each graph at each time independently, and then aligns the time series. This method is suitable for tasks such as predicting links between graphs, since the node representation of each snapshot is independent.
2. Dynamic Triad Embedding: This method uses the concept of triangles to capture the dynamic relationship between nodes. It models how a node is connected to other nodes forming a triangle and reflects changes over time.

Methods that account for time dependence:

1. Dynamic Structural Deep Network Embedding (DSDNE): This method learns the node representation of a graph while considering time-related information. In addition to static network structure information, it incorporates a term to capture changes in the network between time steps.
2. Dynamic Graph Convolutional Networks (DGCN): This method uses convolutional neural networks for dynamic graphs. Convolutional layers are applied at each time step to obtain embeddings that reflect temporal changes in the graph.

Algorithms related to embedding dynamic graphs

Algorithms related to embedding dynamic graphs are described below.

1. Dynamic DeepWalk (DDW): Dynamic DeepWalk applies DeepWalk, a static graph embedding method, to dynamic graphs. For more details on DeepWalk, please refer to “Overview of DeepWalk, Algorithm and Examples of Implementation.

2. Dynamic Node2Vec: Node2Vec is a similar approach to DeepWalk that uses a random walk and skip-gram model to learn node representations, For details of Node2Vec, please refer to “Node2Vec Overview, Algorithm and Example Implementation“.

3. Dynamic Graph Convolutional Networks (DGCN): DGCN is a method using convolutional neural networks (GCN) for dynamic graphs. For details of GCN, please refer to “Overview of Graph Convolutional Neural Networks (GCN), Algorithm and Implementation Examples“.

4. Dynamic Structural Deep Network Embedding (DSDNE): DSDNE is a method that applies SDNE, a static graph embedding method, to dynamic graphs. For more information on SDNE, please refer to “Structural Deep Network Embedding (SDNE) Overview, Algorithms, and Implementation Examples.

5. DynamicTriad: DynamicTriad uses the concept of triangles in dynamic graphs to capture temporal changes in nodes. For more information on DynamicTriad, please refer to “DynamicTriad Overview, Algorithm and Example Implementation.

Examples of applications of embedding in dynamic graphs

Specific applications of dynamic graph embedding are described below.

1. social network analysis: t

Prediction of user behavior: Dynamic graph embeddings are used to model changes in user behavior and relationships based on data from social media platforms and other sources. This allows for the prediction of events such as friend additions, content sharing, and opinion formation.

Community identification: Dynamic graph embedding can be used to analyze user relationships as they change over time and to identify different communities or clusters.

2. optimization of online advertising: 

Analyze user behavior: Dynamic graph embedding can be used to analyze user behavior patterns and trends, optimize the delivery of online advertisements, capture changes in user interests and concerns over time, and deliver advertisements accordingly.

3. analysis of biological networks:

Protein interaction networks: Biological network data can be dynamic in nature, and dynamic graph embedding can be used to analyze changes in protein interaction networks over time to understand protein roles and interaction patterns at different times.

4. prediction of traffic networks:

Traffic Flow Prediction: Dynamic graph embedding can be used to predict trends in urban traffic networks, capturing traffic volumes and vehicle movement patterns to predict congestion and suggest optimal routes.

5. finanCe & econoMy:

Analyzing market fluctuations: financial data such as stock markets and virtual currency markets are time-dependent in nature, and dynamic graph embedding can be used to analyze market trends and correlations to provide useful information to investors and traders.

Example Implementation of Dynamic Graph Embedding

There are several libraries and frameworks for implementing dynamic graph embeddings. In this section, we describe examples of dynamic graph embedding implementations using Python.

1. StellarGraph:

StellarGraph is a Python library for graph machine learning that can easily implement dynamic graph embedding. The following is an example implementation of Dynamic Node2Vec using StellarGraph.

import networkx as nx
from stellargraph import StellarGraph
from stellargraph.data import BiasedRandomWalk
from stellargraph.data import UnsupervisedSampler
from gensim.models import Word2Vec

# 1. create dynamic graphs using NetworkX
G = nx.Graph()
# Graph Construction

# Create a StellarGraph object
graph = StellarGraph.from_networkx(G)

# 3. set up a random walk using BiasedRandomWalk
rw = BiasedRandomWalk(graph)

# 4. sampling with UnsupervisedSampler
nodes = list(graph.nodes())
number_of_walks = 10
length = 80
batch_size = 50
epochs = 3
generator = UnsupervisedSampler(graph, nodes=nodes, length=length, number_of_walks=number_of_walks)
walks = generator.run(batch_size=batch_size, epochs=epochs)

# 5. learning embedding using Word2Vec
model = Word2Vec(walks, vector_size=128, window=5, min_count=0, sg=1, workers=2, epochs=1)

# 6. get embedding of node
node_embeddings = model.wv

2. GraphSAGE:

GraphSAGE described in “GraphSAGE Overview, Algorithm, and Example Implementation“is a method to generate embeddings of nodes from graph data and is applicable to dynamic graphs. The following is an example of implementing Dynamic GraphSAGE using stellargraph.

import networkx as nx
from stellargraph import StellarGraph
from stellargraph.mapper import GraphSAGENodeGenerator
from stellargraph.layer import GraphSAGE
from tensorflow import keras

# 1. create dynamic graphs using NetworkX
G = nx.Graph()
# Graph Construction

# Create a StellarGraph object
graph = StellarGraph.from_networkx(G)

# Define GraphSAGENodeGenerator
batch_size = 50
num_samples = [10, 5]
generator = GraphSAGENodeGenerator(graph, batch_size, num_samples)

# Define GraphSAGE model
layer_sizes = [128, 128]
graphsage = GraphSAGE(
    layer_sizes=layer_sizes, generator=generator, bias=True, dropout=0.5
)

x_inp, x_out = graphsage.in_out_tensors()

# 5. compile the model
prediction = keras.layers.Dense(units=1, activation="sigmoid")(x_out)

model = keras.Model(inputs=x_inp, outputs=prediction)
model.compile(optimizer=keras.optimizers.Adam(lr=1e-3), loss="binary_crossentropy")

# 6. model training
epochs = 5
model.fit(generator.flow(train_subjects.index, train_targets), epochs=epochs, verbose=2)

# 7. get embedding of node
node_ids = graph.nodes()
node_gen = generator.flow(node_ids)
embedding_model = keras.Model(inputs=x_inp, outputs=x_out)
node_embeddings = embedding_model.predict(node_gen)

3. GEMSEC (Graph Embedding with Self Clustering):

GEMSEC is a dynamic graph embedding method with self-clustering. The following is an example implementation using the gemsec package.

from gemsec.algorithm import gemsec
import networkx as nx

# 1. create dynamic graphs using NetworkX
G = nx.Graph()
# Graph Construction

# 2. learning embedding in GEMSEC
model = gemsec(G, t=0, r=1, d=128, context_size=5, num_workers=4)
model.learn_embedding()

# 3. get embedding of node
node_embeddings = model.get_embedding()

Challenges and Countermeasures for Embedding Dynamic Graphs

Embedding dynamic graphs presents several challenges, and there are several measures that can be taken to address these challenges.

1. graph size and scalability:

Challenge: If the dynamic graph is large, computing embeddings is computationally demanding.
Solution:.
Minibatch processing: Split the graph into smaller subgraphs and process them in minibatches to improve computational efficiency.
Sampling: Random sampling from a graph can provide approximate embeddings even for large graphs.

2. robustness to node additions and deletions:

Challenge: In dynamic graphs, embeddings become out of date when new nodes are added or existing nodes are deleted.
Solution:
Re-learning: Re-learn embeddings periodically with new data to reflect the latest information.
Incremental learning: Avoid complete relearning by employing a method of learning with additional new data.

3. modeling changes over time:

Challenge: In dynamic graphs, it is necessary to appropriately capture the changes in relationships and structures among nodes over time.
Solution:
Dynamic graph models: Use models that account for time variation (e.g., DynamicTriad, Dynamic Node2Vec).
Incorporate time-series information: Incorporate temporal characteristics into node embeddings to reflect temporal changes.

4. difficulty in evaluation:

Challenge: Evaluation of embeddings in dynamic graphs is more difficult than in static graphs and requires appropriate evaluation metrics.
Solution:
Task-oriented evaluation: Evaluate embedding quality in terms of performance on specific tasks.
Predictive tasks: evaluate embeddings on tasks such as link prediction, graph clustering, node classification, etc.

5. dealing with missing data and noise:

Challenge: Real-world dynamic graph data contains missing and noisy data.
Solution:
Dealing with missing values: Adopt appropriate completion methods for missing values.
Noise reduction: Use pre-processing methods to remove noise.

6. domain specific challenges:

Challenges: Different domains may have different properties and problems with dynamic graphs.
Solution:
Incorporate domain knowledge: work with domain experts to address domain-specific challenges.
Domain Adaptation: Adapt embeddings learned in other domains to a specific domain by transfer learning described in “Overview of Transfer Learning and Examples of Algorithms and Implementations“.

Reference Information and Reference Books

For more information on graph data, see “Graph Data Processing Algorithms and Applications to Machine Learning/Artificial Intelligence Tasks. Also see “Knowledge Information Processing Techniques” for details specific to knowledge graphs. For more information on deep learning in general, see “About Deep Learning.

Reference book is

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

“Graph Neural Networks: Foundations, Frontiers, and Applications“等がある。

“Introduction to Graph Neural Networks“

“Graph Neural Networks in Action“