Louvain Community Detection
Overview
Louvain algorithm detects communities by optimizing modularity — the fraction of edges within communities minus expected fraction if edges were random. A greedy, hierarchical algorithm that runs in O(n log n) for sparse graphs. Produces a hierarchy of communities at multiple resolutions.
When to Use
Trigger conditions:
- Discovering natural groupings in social, organizational, or interaction networks
- Segmenting users/customers by behavioral similarity
- Analyzing modular structure of complex networks
When NOT to use:
- For overlapping communities (use DEMON or BigCLAM)
- When communities are pre-defined and you're classifying nodes (use label propagation)
Algorithm
IRON LAW: Modularity Has a RESOLUTION LIMIT Louvain optimizes modularity, which has a known resolution limit (Fortunato & Barthélemy, 2007): it cannot detect communities smaller than √(2E) where E = total edges. In large networks, small but real communities may be merged. Use multi-resolution methods or Leiden algorithm (improved Louvain) for better results.
Phase 1: Input Validation
Build undirected weighted graph from interaction data. Edge weights represent interaction strength (frequency, duration, volume). Gate: Graph loaded, no isolated nodes (or decide how to handle them).
Phase 2: Core Algorithm
Phase 1 — Local moves:
- Assign each node to its own community
- For each node, compute modularity gain of moving to each neighbor's community
- Move node to community with maximum positive gain
- Repeat until no beneficial moves remain
Phase 2 — Aggregation: 5. Build new graph where nodes = communities, edges = sum of inter-community edges 6. Repeat Phase 1 on the aggregated graph 7. Continue until modularity stops improving
Phase 3: Verification
Check: modularity Q > 0 (non-trivial partitioning), community sizes are reasonable (not one giant + many singletons), manual inspection of sample communities. Gate: Modularity positive, community sizes follow power-law-like distribution.
Phase 4: Output
Return community assignments with modularity score.
Output Format
json{ "communities": [{"id": 0, "size": 45, "top_members": ["Alice", "Bob"], "internal_density": 0.35}], "summary": {"num_communities": 12, "modularity": 0.65, "largest": 120, "smallest": 5}, "metadata": {"algorithm": "louvain", "nodes": 500, "edges": 2000} }
Examples
Sample I/O
Input: Email network of 200 employees, weighted by email frequency Expected: Communities roughly corresponding to departments/teams, modularity ~0.5-0.7.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| Complete graph | One community or random split | No modular structure |
| Disconnected components | Each component = community | Natural separation |
| Weighted vs unweighted | Different communities | Weights change modularity calculation |
Gotchas
- Non-deterministic: Node processing order affects results. Run multiple times and select the partition with highest modularity, or use Leiden algorithm (more stable).
- Resolution parameter: Standard Louvain uses γ=1 in modularity. Varying γ reveals communities at different scales. γ>1 finds smaller communities; γ<1 finds larger ones.
- Leiden > Louvain: Louvain can produce badly connected communities (communities where removing one node disconnects them). Leiden algorithm fixes this guarantee.
- Temporal stability: In dynamic networks, community assignments can change drastically between snapshots even when the network changes minimally. Use temporal smoothing.
- Interpretation: Community detection finds structure, but interpreting WHY nodes cluster requires domain knowledge. Don't over-interpret automatically detected communities.
References
- For Leiden algorithm (improved Louvain), see
references/leiden.md - For multi-resolution community detection, see
references/multi-resolution.md
