Implement Community Detection (Louvain/Leiden) for Graphs

[noahgift]

Problem Statement

Community detection algorithms identify densely connected groups (communities) in networks. Currently missing from aprender's graph module.

Use Cases:

• Social network analysis (find friend groups)
• Protein interaction networks (functional modules)
• Citation networks (research topics)
• Web page clustering
• Recommendation systems

Algorithms:

• Louvain: Fast, greedy modularity optimization
• Leiden: Improved version (fixes disconnected communities)

Proposed Solution

Implement Louvain and Leiden algorithms in the graph module with EXTREME TDD.

Algorithm (Louvain)

Modularity Optimization:

• Measures density of edges within communities vs between
• Q = (1/2m) Σ[A_ij - k_i*k_j/2m] δ(c_i, c_j)

Steps:

1. Phase 1: Each node in own community
   • For each node: try moving to neighbor communities
   • Accept move if modularity increases
   • Repeat until no improvement
2. Phase 2: Build new graph where nodes = communities
   • Repeat Phase 1 on new graph
3. Continue until modularity converges

Implementation

API Design:

// In src/graph/mod.rs

pub struct Community {
    pub nodes: Vec<NodeId>,
    pub modularity: f64,
}

impl Graph {
    pub fn louvain(&self) -> Vec<Community>;
    pub fn leiden(&self, resolution: f64) -> Vec<Community>;
    pub fn modularity(&self, communities: &[Vec<NodeId>]) -> f64;
}

Success Criteria

• ✅ Louvain algorithm implementation
• ✅ Leiden algorithm implementation (optional, better quality)
• ✅ Modularity computation
• ✅ 15+ tests (including known community structures)
• ✅ Zero clippy warnings
• ✅ Example: examples/community_detection.rs

Estimated Effort

Timeline: 4-5 days
Complexity: Medium-High (modularity optimization, hierarchical structure)

[noahgift]

✅ Completed: Community Detection with Louvain algorithm implemented with 16 comprehensive tests, example, and case study documentation. All quality gates passing.