🧠 AI Exploration #9: Leiden Clustering Explained

Leiden Clustering is a powerful graph-based clustering algorithm used to discover communities in complex networks.

Unlike K-Means, Leiden does not assume spherical clusters. Instead, it works on a graph of relationships between samples and groups together nodes that are densely connected.

Today, Leiden is widely used in:

• Semantic embedding clustering
• Single-cell biology
• Recommendation systems
• Social network analysis
• Large-scale AI retrieval systems

🧠 How Leiden Clustering Works

Leiden operates on a graph:

• Each sample becomes a node
• Similar samples are connected by edges
• Edge weights represent similarity strength

The algorithm then searches for communities where:

• connections inside the cluster are strong
• connections between clusters are weak

Typical Pipeline

A common modern AI clustering pipeline looks like this:

Raw Text
   ↓
Embedding Model
   ↓
Vector Embeddings
   ↓
k-Nearest Neighbor Graph
   ↓
Leiden Clustering

For example:

• Sentence embeddings from BGE or OpenAI
• kNN graph using cosine similarity
• Leiden for community detection

🧮 Mathematical Intuition

Leiden optimizes a quantity called modularity (or a related objective).

The idea is:

A good cluster should contain more internal connections than expected by random chance.

Graph Representation

Let:

• $G = (V, E)$
• $V$ = set of nodes
• $E$ = weighted edges

Each edge has weight:

representing similarity between node $i$ and node $j$.

Modularity Objective

A simplified modularity formulation is:

Where:

• $A_{ij}$ = edge weight between nodes
• $k_i$ = total degree of node $i$
• $m$ = total graph weight
• $c_i$ = cluster assignment
• $\delta(c_i, c_j) = 1$ if nodes belong to same cluster

Higher modularity means stronger community structure.

🔍 Why Leiden Is Better Than Louvain

Leiden is actually an improvement over the older Louvain algorithm.

Louvain Problem

Louvain may produce:

• disconnected communities
• poorly connected clusters
• unstable partitions

Leiden Improvement

Leiden adds:

• refinement steps
• connectivity guarantees
• better optimization stability

As a result:

• clusters are more coherent
• results are more robust
• convergence is usually faster

🎯 When to Use Leiden

Leiden works especially well when:

• Using embedding vectors
• Clustering semantic text
• Discovering graph communities
• Cluster shapes are non-spherical
• Dataset structure is complex

It is often better than K-Means for modern embedding spaces.

✅ Advantages and Disadvantages

✅ Pros

• Excellent for embedding clustering
• Handles irregular cluster structure
• No need to assume spherical clusters
• Scales well to large graphs
• Produces well-connected communities

❌ Cons

• Requires graph construction
• Sensitive to graph quality
• kNN and resolution tuning matter
• Harder to interpret mathematically than K-Means

🧪 Code Example: Leiden Clustering with Sentence Embeddings

import numpy as np
import igraph as ig
import leidenalg

from sentence_transformers import SentenceTransformer
from sklearn.neighbors import NearestNeighbors

texts = [
    "database connection failed",
    "unable to connect to database",
    "gpu memory overflow",
    "cuda out of memory",
    "payment transaction failed",
]

# Generate embeddings
model = SentenceTransformer("BAAI/bge-small-en-v1.5")

embeddings = model.encode(
    texts,
    normalize_embeddings=True,
)

# Build kNN graph
k = 2

nn = NearestNeighbors(
    n_neighbors=k + 1,
    metric="cosine",
)

nn.fit(embeddings)

distances, indices = nn.kneighbors(embeddings)

edges = []
weights = []

for source_idx, (row_distances, row_indices) in enumerate(
    zip(distances, indices)
):
    for distance, target_idx in zip(row_distances, row_indices):

        if source_idx == target_idx:
            continue

        similarity = 1.0 - distance

        edges.append((source_idx, target_idx))
        weights.append(similarity)

# Create graph
graph = ig.Graph(
    n=len(texts),
    edges=edges,
    directed=False,
)

graph.es["weight"] = weights

# Run Leiden
partition = leidenalg.find_partition(
    graph,
    leidenalg.RBConfigurationVertexPartition,
    weights=graph.es["weight"],
    resolution_parameter=0.5,
)

labels = partition.membership

for text, label in zip(texts, labels):
    print(label, text)

This example clusters semantically similar log messages together using:

• BGE embeddings
• cosine similarity
• kNN graph construction
• Leiden community detection

📊 Understanding Important Parameters

k (kNN neighbors)

Controls graph connectivity.

• Smaller k → fragmented graph → more clusters

• Larger k → denser graph → fewer clusters

resolution_parameter

Controls cluster granularity.

• Lower resolution → larger clusters

• Higher resolution → smaller clusters

This is one of the most important tuning parameters in Leiden.

🔬 t-SNE Visualization

A common practice is:

1. Generate embeddings
2. Run Leiden clustering
3. Use t-SNE only for visualization

Important:

t-SNE is NOT the clustering algorithm itself.

It only projects high-dimensional embeddings into 2D for visualization.

🔚 Recap

Leiden Clustering is one of the most effective modern clustering techniques for embedding-based AI systems.

Instead of assuming geometric cluster shapes, Leiden discovers communities through graph connectivity - making it especially powerful for:

• semantic search
• observability systems
• recommendation engines
• retrieval pipelines
• large-scale representation learning

🔜 Coming Next

Next in this graph clustering subseries:

Louvain vs. Leiden — understanding community detection quality, modularity optimization, and clustering stability.

Stay curious and keep exploring 👇