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README.md

@@ -10,156 +10,136 @@ tags:
   - mathlib
   - dependency-graph
   - formal-verification
+  - network-analysis
 size_categories:
   - 100K<n<1M
 ---
 
-# MathlibGraph: Mathlib Dependency Graph Dataset
+# MathlibGraph: The Multinetwork of Mathlib
 
-Dependency graph of [Mathlib](https://github.com/leanprover-community/mathlib4) (commit [`534cf0b`](https://github.com/leanprover-community/mathlib4/commit/534cf0b), 2 Feb 2026), the largest formal mathematics library for Lean 4. Lean version: `v4.28.0-rc1`.
+Dependency graph of [Mathlib](https://github.com/leanprover-community/mathlib4) (commit [`534cf0b`](https://github.com/leanprover-community/mathlib4/commit/534cf0b), 2 Feb 2026), the largest formal mathematics library for Lean 4 (`v4.28.0-rc1`).
+
+Three dependency layers (declarations, modules, namespaces) with precomputed network metrics.
+
+## Quick Stats
+
+| | Declarations | Modules | Namespaces (k=2) |
+|---|---|---|---|
+| Nodes | 308,129 | 7,564 | 10,097 |
+| Edges | 8,436,366 | 20,881 | 332,081 |
+| DAG depth | 83 | 154 | 7 |
+| Louvain modularity | 0.478 | 0.610 | 0.270 |
+| Synthesized edges | 74.2% | — | — |
+| Cross-namespace edges | — | — | 85.8% |
 
 ## Files
 
-### Core data (v1)
+### Raw Graph Data
 
 | File | Rows | Description |
 |------|------|-------------|
-| `mathlib_nodes.csv` | 317,655 (308,129 unique) | Mathlib declarations (name, kind, module) |
-| `mathlib_edges.csv` | 8,436,366 | Mathlib dependencies (source, target, is_explicit, is_simplifier) |
-| `nodes.csv` | 633,364 | Full environment (Lean + Std + Mathlib) |
+| `mathlib_nodes.csv` | 317,655 | Declarations before deduplication (name, kind, module) |
+| `mathlib_edges.csv` | 8,436,366 | Declaration dependencies (source, target, is_explicit, is_simplifier) |
+| `nodes.csv` | 633,364 | Full environment including Lean core and Std |
 | `edges.csv` | 10,889,011 | Full environment edges |
 | `mechanisms.ndjson` | — | Lean language mechanism extractions |
-| `tactic_usage.ndjson` | — | Tactic usage profiles |
+| `tactic_usage.ndjson` | — | Per-declaration tactic usage profiles |
 
-### Enriched data (v2)
+### Enriched Data (v2)
 
-| File | Rows | Description |
-|------|------|-------------|
-| `v2/mathlib_nodes_enriched.csv` | 308,129 | Deduplicated nodes with 8 precomputed metric columns |
-| `v2/mathlib_modules.csv` | 7,564 | Per-module (source file) metrics |
-| `v2/mathlib_namespaces_k2.csv` | 10,097 | Per-namespace (depth-2) metrics |
-| `v2/mathlib_summary.json` | — | Headline statistics for all three graph levels |
-
-> **Note:** `mathlib_nodes.csv` contains 9,526 duplicate name rows from `@[to_additive]` mirroring. The v2 enriched file is deduplicated to **308,129 unique declarations**.
-
-## Statistics (308,129 unique Mathlib declarations)
-
-| Kind | Count | % |
-|------|-------|---|
-| theorem | 251,642 | 79.2% |
-| definition | 50,156 | 15.8% |
-| abbrev | 6,784 | 2.1% |
-| constructor | 4,755 | 1.5% |
-| inductive | 3,813 | 1.2% |
-| opaque | 499 | 0.2% |
-| quotient | 3 | <0.1% |
-| axiom | 3 | <0.1% |
+| File | Rows | Columns | Description |
+|------|------|---------|-------------|
+| `v2/mathlib_nodes_enriched.csv` | 308,129 | 11 | Deduplicated declarations with network metrics |
+| `v2/mathlib_modules.csv` | 7,564 | 10 | Per-module (source file) metrics |
+| `v2/mathlib_namespaces_k2.csv` | 10,097 | 9 | Per-namespace (depth-2) metrics |
+| `v2/mathlib_summary.json` | — | — | Headline statistics for all three graph levels |
 
 ## Quick Start
 
 ```python
 from datasets import load_dataset
 
-# Load enriched nodes (v2, deduplicated, with metrics)
 nodes = load_dataset("MathNetwork/MathlibGraph",
                      data_files="v2/mathlib_nodes_enriched.csv", split="train").to_pandas()
 edges = load_dataset("MathNetwork/MathlibGraph",
                      data_files="mathlib_edges.csv", split="train").to_pandas()
 
 print(f"Declarations: {len(nodes):,}, Edges: {len(edges):,}")
-# Declarations: 308,129, Edges: 8,436,366
-
-# Top 10 by PageRank
-print(nodes.nlargest(10, "pagerank")[["name", "kind", "in_degree", "pagerank"]])
-
-# Load module-level data
-modules = load_dataset("MathNetwork/MathlibGraph",
-                       data_files="v2/mathlib_modules.csv", split="train").to_pandas()
-print(f"Modules: {len(modules):,}")
+print(nodes.nlargest(5, "pagerank")[["name", "kind", "in_degree", "pagerank"]])
 ```
 
 ## Schema
 
-### mathlib_nodes.csv (v1)
-
-| Column | Type | Description |
-|--------|------|-------------|
-| name | string | Fully qualified declaration name |
-| kind | string | `theorem`, `definition`, `abbrev`, `inductive`, `constructor`, `opaque`, `axiom`, `quotient` |
-| module | string | Parent namespace of the declaration |
-
-### mathlib_edges.csv
-
-| Column | Type | Description |
-|--------|------|-------------|
-| source | string | Dependent declaration |
-| target | string | Dependency (premise) |
-| is_explicit | bool | Appears in explicit arguments (25.8% of edges) |
-| is_simplifier | bool | Used by simplifier |
-
 ### v2/mathlib_nodes_enriched.csv
 
-All columns from `mathlib_nodes.csv` plus:
-
 | Column | Type | Description |
 |--------|------|-------------|
-| namespace_depth2 | string | Depth-2 namespace (e.g., `Mathlib.Algebra`) |
-| namespace_depth3 | string | Depth-3 namespace (e.g., `Mathlib.Algebra.Group`) |
-| in_degree | int | Number of declarations that depend on this one |
-| out_degree | int | Number of declarations this one depends on |
-| pagerank | float | PageRank score (alpha=0.85) |
+| name | str | Fully qualified declaration name |
+| kind | str | One of: theorem, definition, abbrev, inductive, constructor, opaque, axiom, quotient |
+| module | str | Parent namespace of the declaration (null for 30,944 Lean core declarations) |
+| namespace_depth2 | str | First 2 dot-separated name components (e.g., `Mathlib.Algebra`) |
+| namespace_depth3 | str | First 3 dot-separated name components |
+| in_degree | int | Declarations that depend on this one |
+| out_degree | int | Declarations this one depends on |
+| pagerank | float | PageRank (alpha=0.85) |
 | betweenness | float | Betweenness centrality (sampled, k=500, seed=42) |
-| community_id | int | Louvain community ID (resolution=1.0, seed=42) |
-| dag_layer | int | Topological depth in DAG (-1 if in a cycle; 5,732 nodes) |
+| community_id | int | Louvain community assignment |
+| dag_layer | int | Topological depth; -1 for 5,732 nodes in cycles |
 
 ### v2/mathlib_modules.csv
 
-One row per Mathlib source file (7,564 modules).
-
 | Column | Type | Description |
 |--------|------|-------------|
-| module | string | Dotted module name (e.g., `Mathlib.Algebra.Group.Defs`) |
-| decl_count | int | Declarations in this module (from jixia extraction, 235K coverage) |
-| in_degree | int | Modules that import this module |
-| out_degree | int | Modules this module imports |
+| module | str | Dotted module name (e.g., `Mathlib.Algebra.Group.Defs`) |
+| decl_count | int | Declarations in this module (235K jixia coverage) |
+| in_degree | int | Modules that import this one |
+| out_degree | int | Modules this one imports |
 | pagerank | float | PageRank on module import graph |
 | betweenness | float | Betweenness centrality (exact, no sampling) |
 | dag_layer | int | Topological depth in module DAG |
-| community_id | int | Louvain community ID |
-| cohesion | float | internal_edges / (internal + external) |
+| community_id | int | Louvain community assignment |
+| cohesion | float | Fraction of edges that stay within the module |
 | import_utilization_median | float | Median fraction of imported declarations actually used |
 
 ### v2/mathlib_namespaces_k2.csv
 
-One row per depth-2 namespace (10,097 namespaces).
-
 | Column | Type | Description |
 |--------|------|-------------|
-| namespace | string | Depth-2 namespace (e.g., `Mathlib.Algebra`) |
+| namespace | str | Depth-2 namespace (e.g., `Mathlib.Algebra`) |
 | decl_count | int | Declarations in this namespace |
 | in_degree | int | Unweighted in-degree in namespace graph |
-| out_degree | int | Unweighted out-degree in namespace graph |
-| edge_weight_sum | int | Weighted degree (sum of declaration-level edges) |
-| pagerank | float | PageRank (weighted, alpha=0.85) |
-| betweenness | float | Betweenness centrality (k=300, seed=42, weighted) |
-| community_id | int | Louvain community ID (weighted undirected) |
-| cross_ns_ratio | float | Fraction of this namespace's edges that cross namespace boundaries |
+| out_degree | int | Unweighted out-degree |
+| edge_weight_sum | int | Sum of declaration-level edges involving this namespace |
+| pagerank | float | Weighted PageRank (alpha=0.85) |
+| betweenness | float | Weighted betweenness (k=300, seed=42) |
+| community_id | int | Louvain community on weighted undirected graph |
+| cross_ns_ratio | float | Fraction of edges crossing namespace boundaries |
 
 ### v2/mathlib_summary.json
 
-Headline statistics including: snapshot metadata (commit, date, Lean version), declaration/module/namespace graph metrics (node/edge counts, modularity, NMI, DAG depth, power-law exponents), and edge decomposition ratios.
+Top-level keys: `snapshot` (commit, date, Lean version), `declaration_graph`, `module_graph`, `namespace_graph_k2` (node/edge counts, modularity, NMI, DAG depth, power-law exponents), `edge_decompositions` (proof-only/statement-only ratios), `kind_distribution`.
 
-## Extraction Pipeline
+## Methodology
 
-- **Nodes**: [lean4export](https://github.com/leanprover/lean4export)
-- **Edges**: [lean-training-data premises](https://github.com/kim-em/lean-training-data)
-- **Module graph**: [importGraph](https://github.com/leanprover-community/importGraph)
-- **Enrichment**: NetworkX (PageRank, betweenness, Louvain, DAG layers)
-- **Code**: [MathlibGraph on GitHub](https://github.com/MathNetwork/MathlibGraph)
+- **Extraction**: lean4export (nodes) + lean-training-data (edges) + importGraph (module graph) + jixia (declaration metadata)
+- **Deduplication**: by name, 317,655 to 308,129 rows (9,526 `@[to_additive]` mirrors removed)
+- **Self-loops**: 4,755 constructor self-references filtered from graph construction
+- **PageRank**: alpha=0.85, max_iter=100, tol=1e-6
+- **Betweenness**: declaration k=500, namespace k=300, module exact; all with seed=42
+- **Communities**: Louvain (python-louvain), resolution=1.0, random_state=42, on undirected projection
+- **DAG layers**: Kahn's algorithm; 5,732 declaration nodes in cycles assigned layer=-1; namespace graph condensed via SCC
 
 ## Citation
 
-Part of the [MathlibGraph](https://github.com/MathNetwork/MathlibGraph) project — network analysis of formalized mathematics.
+```bibtex
+@inproceedings{mathlibgraph2026,
+  title     = {The Multinetwork of Mathlib},
+  author    = {Li, Xinze and Shafto, Patrick},
+  booktitle = {NeurIPS 2026 Datasets and Benchmarks Track},
+  year      = {2026},
+  note      = {Under review}
+}
+```
 
 ## License