RFC-004: Graph-Based Semantic Regions

This RFC proposes using the HNSW graph’s hierarchical structure to define semantic regions — natural clusters derived from graph topology — transforming noisy per-post trajectories into smooth, interpretable temporal signals.

Problem

Individual post embeddings are too noisy for temporal analysis. A depressed user still writes about sports, food, weather — topics that dominate the embedding trajectory. Direct analysis on D=384 raw embeddings yields:

Metric	Result (eRisk)
Velocity Cohen’s d	-0.03 (no discrimination)
PELT change points	0 detected
Hurst p-value	0.25 (not significant)

Key Insight

The HNSW graph already is a hierarchical clustering:

• Level 0: all nodes
• Level L: ~N/M^L “hub” nodes (natural centroids)
• Choosing level controls granularity

Solution

1. Extract regions from HNSW level L (e.g., ~30-500 regions)
2. Assign each post to nearest region hub (O(log N) via graph descent)
3. Compute temporal distribution per user over regions in sliding windows
4. Smooth with EMA: $s_t = \alpha \cdot x_t + (1-\alpha) \cdot s_{t-1}$
5. Apply CVX analytics (velocity, CPD, Hurst) on smooth K-dimensional signal

Result

Instead of “dim_127 changed by 0.3”, we get: “user shifted from 40% social to 60% emotional distress over 3 weeks”.

New API

regions = index.regions(level=2)            # graph-derived clusters
traj = index.region_trajectory(uid, level=2, window_days=7, alpha=0.3)
members = index.region_members(region_id=7)  # single region
all_members = index.region_assignments(level=2)  # all regions in one O(N) pass

Phases

Phase	Scope	Effort
1	Graph region extraction (nodes_at_level, assign_region)	Low
2	Region trajectory computation (sliding window + EMA)	Medium
3	Region inspection (region_members, region_assignments)	Low
4	Python bindings	Low
5	Tutorial B1 rewrite with real results	Medium

See full RFC in design/CVX_RFC_004_Graph_Semantic_Regions.md.