Impara Node Similarity and Clustering | Graph-Based Machine Learning Tasks

Node embeddings turn graph nodes into vectors, allowing you to use mathematical operations to compare nodes and discover patterns. Measuring node similarity with embeddings is typically done using a similarity metric such as cosine similarity. This approach captures how "close" or "related" two nodes are based on their feature representations, rather than just direct graph connections. When you have embeddings for all nodes, you can build a similarity matrix that quantifies the relationships throughout the graph.

Clustering is another powerful tool enabled by embeddings. By grouping nodes with similar embeddings, clustering methods like k-means can reveal communities or modules within the graph. These clusters might correspond to groups of users with similar interests in a social network, or proteins with related functions in a biological network. Unlike node similarity, which focuses on pairs of nodes, clustering considers the global structure and seeks to partition the graph into meaningful subsets.


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import numpy as np
from sklearn.cluster import KMeans

# Example node embeddings (each row is a node embedding)
embeddings = np.array([
    [0.1, 0.2, 0.9],
    [0.2, 0.1, 0.8],
    [0.9, 0.8, 0.2],
    [0.8, 0.9, 0.1]
])

# Compute pairwise cosine similarity
def cosine_similarity_matrix(X):
    norm = np.linalg.norm(X, axis=1, keepdims=True)
    X_normalized = X / norm
    return np.dot(X_normalized, X_normalized.T)

similarity_matrix = cosine_similarity_matrix(embeddings)
print("Pairwise Cosine Similarity Matrix:")
print(similarity_matrix)

# Perform k-means clustering (e.g., 2 clusters)
kmeans = KMeans(n_clusters=2, random_state=42, n_init=10)
labels = kmeans.fit_predict(embeddings)
print("Cluster assignments for each node:")
print(labels)

Node similarity

Measures how alike two specific nodes are, based on their embeddings; it answers the question, "Are these two nodes similar?"

Node clustering

Groups all nodes into subsets (clusters) so that nodes in the same cluster are more similar to each other than to those in other clusters; it answers, "Which nodes form natural groups or communities?"

1. What is the goal of clustering nodes in a graph?

2. How does cosine similarity help in finding similar nodes?

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              1234567891011121314151617181920212223242526
            
import numpy as np
from sklearn.cluster import KMeans

# Example node embeddings (each row is a node embedding)
embeddings = np.array([
    [0.1, 0.2, 0.9],
    [0.2, 0.1, 0.8],
    [0.9, 0.8, 0.2],
    [0.8, 0.9, 0.1]
])

# Compute pairwise cosine similarity
def cosine_similarity_matrix(X):
    norm = np.linalg.norm(X, axis=1, keepdims=True)
    X_normalized = X / norm
    return np.dot(X_normalized, X_normalized.T)

similarity_matrix = cosine_similarity_matrix(embeddings)
print("Pairwise Cosine Similarity Matrix:")
print(similarity_matrix)

# Perform k-means clustering (e.g., 2 clusters)
kmeans = KMeans(n_clusters=2, random_state=42, n_init=10)
labels = kmeans.fit_predict(embeddings)
print("Cluster assignments for each node:")
print(labels)