fix load + add another example

README.md

@@ -68,7 +68,7 @@ print(df)
 print(kc.dump_graph())       # Turtle string
 ```
 
-See [`examples/`](examples/) for 10 runnable examples covering all features below.
+See [`examples/`](examples/) for 11 runnable examples covering all features below.
 
 ## Topological queries
 
@@ -132,6 +132,23 @@ chi = euler_characteristic(kc)     # V - E + F
 pr = edge_pagerank(kc, "e1")       # personalized edge PageRank vector
 ```
 
+## Local partitioning
+
+Find clusters via diffusion — spread probability from a seed and sweep to find natural bottlenecks:
+
+```python
+from knowledgecomplex.analysis import local_partition, edge_local_partition
+
+# Vertex clusters via PageRank or heat kernel diffusion
+cut = local_partition(kc, seed="alice", method="pagerank")
+cut.vertices       # vertex IDs on the small side
+cut.conductance    # lower = cleaner partition
+
+# Edge clusters via Hodge Laplacian diffusion
+edge_cut = edge_local_partition(kc, seed_edge="e1", method="hodge_pagerank")
+edge_cut.edges     # relationship cluster around e1
+```
+
 ## Filtrations and time-varying complexes
 
 Filtrations model strictly growing subcomplexes. Diffs model arbitrary add/remove sequences:

docs/index.md

@@ -64,7 +64,7 @@ from knowledgecomplex.ontologies import operations, brand, research
 sb = brand.schema()   # audience/theme with resonance, interplay, overlap
 ```
 
-See the [examples/](https://github.com/blockscience/knowledgecomplex/tree/main/examples) directory for 10 runnable examples.
+See the [examples/](https://github.com/blockscience/knowledgecomplex/tree/main/examples) directory for 11 runnable examples.
 
 ## API Reference
 

docs/tutorial.md

@@ -0,0 +1,341 @@
+# Tutorial
+
+A progressive walkthrough of knowledgecomplex, from schema definition through algebraic topology.
+
+## 1. Define a schema
+
+A schema declares vertex, edge, and face types with attributes. The `SchemaBuilder` generates OWL and SHACL automatically.
+
+```python
+from knowledgecomplex import SchemaBuilder, vocab, text
+
+sb = SchemaBuilder(namespace="vv")
+
+# Vertex types (subclass of kc:Vertex)
+sb.add_vertex_type("requirement", attributes={"title": text()})
+sb.add_vertex_type("test_case",   attributes={"title": text()})
+
+# Edge type with controlled vocabulary (enforced via sh:in)
+sb.add_edge_type("verifies", attributes={
+    "status": vocab("passing", "failing", "pending"),
+})
+
+# Face type
+sb.add_face_type("coverage")
+```
+
+### Attribute descriptors
+
+| Descriptor | What it generates | Example |
+|---|---|---|
+| `text()` | `xsd:string`, required, single-valued | `title: text()` |
+| `text(required=False)` | `xsd:string`, optional | `notes: text(required=False)` |
+| `text(multiple=True)` | `xsd:string`, required, multi-valued | `tags: text(multiple=True)` |
+| `vocab("a", "b")` | `sh:in ("a" "b")`, required, single-valued | `status: vocab("pass", "fail")` |
+
+### Type inheritance and binding
+
+Types can inherit from other user-defined types. Child types can bind inherited attributes to fixed values:
+
+```python
+sb.add_vertex_type("document", attributes={"title": text(), "category": text()})
+sb.add_vertex_type("specification", parent="document",
+                   attributes={"format": text()},
+                   bind={"category": "structural"})
+```
+
+### Introspection
+
+```python
+sb.describe_type("specification")
+# {'name': 'specification', 'kind': 'vertex', 'parent': 'document',
+#  'own_attributes': {'format': text()},
+#  'inherited_attributes': {'title': text(), 'category': text()},
+#  'all_attributes': {'title': text(), 'category': text(), 'format': text()},
+#  'bound': {'category': 'structural'}}
+
+sb.type_names(kind="vertex")   # ['document', 'specification']
+```
+
+## 2. Build a complex
+
+A `KnowledgeComplex` manages instances. Every write triggers SHACL verification — the graph is always in a valid state.
+
+```python
+from knowledgecomplex import KnowledgeComplex
+
+kc = KnowledgeComplex(schema=sb)
+
+# Vertices have no boundary — always valid
+kc.add_vertex("req-001", type="requirement", title="Boot time < 5s")
+kc.add_vertex("tc-001",  type="test_case",   title="Boot smoke test")
+kc.add_vertex("tc-002",  type="test_case",   title="Boot regression")
+
+# Edges need their boundary vertices to already exist (slice rule)
+kc.add_edge("ver-001", type="verifies",
+            vertices={"req-001", "tc-001"}, status="passing")
+kc.add_edge("ver-002", type="verifies",
+            vertices={"req-001", "tc-002"}, status="pending")
+kc.add_edge("ver-003", type="verifies",
+            vertices={"tc-001", "tc-002"}, status="passing")
+
+# Faces need 3 boundary edges forming a closed triangle
+kc.add_face("cov-001", type="coverage",
+            boundary=["ver-001", "ver-002", "ver-003"])
+```
+
+### What gets enforced
+
+| Constraint | When | What happens |
+|---|---|---|
+| Type must be registered | Before RDF assertions | `ValidationError` |
+| Boundary cardinality (2 for edges, 3 for faces) | Before SHACL | `ValueError` |
+| Boundary elements must exist in complex (slice rule) | SHACL on write | `ValidationError` + rollback |
+| Vocab values must be in allowed set | SHACL on write | `ValidationError` + rollback |
+| Face boundary edges must form closed triangle | SHACL on write | `ValidationError` + rollback |
+
+### Element handles
+
+```python
+elem = kc.element("req-001")
+elem.id      # "req-001"
+elem.type    # "requirement"
+elem.attrs   # {"title": "Boot time < 5s"}
+
+kc.element_ids(type="test_case")   # ["tc-001", "tc-002"]
+kc.elements(type="test_case")      # [Element('tc-001', ...), Element('tc-002', ...)]
+```
+
+## 3. Topological queries
+
+Every query returns `set[str]` for natural set algebra. All accept an optional `type=` filter.
+
+```python
+# Boundary operator ∂
+kc.boundary("ver-001")     # {'req-001', 'tc-001'}  (edge → vertices)
+kc.boundary("cov-001")     # {'ver-001', 'ver-002', 'ver-003'}  (face → edges)
+kc.boundary("req-001")     # set()  (vertex → empty)
+
+# Coboundary (inverse boundary)
+kc.coboundary("req-001")   # {'ver-001', 'ver-002'}  (vertex → incident edges)
+
+# Star: all simplices containing σ as a face
+kc.star("req-001")         # req-001 + incident edges + incident faces
+
+# Closure: smallest subcomplex containing σ
+kc.closure("cov-001")      # cov-001 + 3 edges + 3 vertices
+
+# Link: Cl(St(σ)) \ St(σ)
+kc.link("req-001")
+
+# Skeleton: elements up to dimension k
+kc.skeleton(0)   # vertices only
+kc.skeleton(1)   # vertices + edges
+
+# Degree
+kc.degree("req-001")   # 2
+
+# Subcomplex check
+kc.is_subcomplex({"req-001", "tc-001", "ver-001"})   # True
+kc.is_subcomplex({"ver-001"})                         # False (missing vertices)
+
+# Set algebra composes naturally
+shared = kc.star("req-001") & kc.star("tc-001")
+```
+
+## 4. Local partitioning
+
+The topological queries above use combinatorial adjacency — boundary, star, and closure walk the simplicial structure directly. Local partitioning uses **diffusion** instead: spread probability from a seed and sweep the result to find a natural cluster boundary. This finds structure that combinatorial queries miss.
+
+Requires `pip install knowledgecomplex[analysis]`.
+
+### Graph partitioning (vertex clusters)
+
+Diffuse from a seed vertex using personalized PageRank or the heat kernel, then sweep the resulting distribution to find a cut with low conductance:
+
+```python
+from knowledgecomplex.analysis import (
+    approximate_pagerank, heat_kernel_pagerank,
+    sweep_cut, local_partition,
+)
+
+# Approximate PageRank: push-based diffusion (Andersen-Chung-Lang)
+p, r = approximate_pagerank(kc, seed="req-001", alpha=0.15)
+# p is a sparse dict of vertex → probability; more mass near seed
+
+# Heat kernel PageRank: exponential diffusion (Fan Chung)
+rho = heat_kernel_pagerank(kc, seed="req-001", t=5.0)
+# t controls locality: small t = tight cluster, large t = broad spread
+
+# Sweep either distribution to find a low-conductance cut
+cut = sweep_cut(kc, p)
+cut.vertices      # set of vertex IDs on the small side
+cut.conductance   # Cheeger ratio — lower means cleaner partition
+
+# Or use local_partition for the full pipeline in one call
+cut = local_partition(kc, seed="req-001", method="pagerank")
+cut = local_partition(kc, seed="req-001", method="heat_kernel")
+```
+
+### Edge partitioning (simplicial clusters)
+
+The simplicial version replaces the graph Laplacian with the **Hodge Laplacian** on edges. Instead of partitioning vertices, it partitions edges — finding clusters of relationships:
+
+```python
+from knowledgecomplex.analysis import edge_local_partition
+
+# Hodge PageRank: (βI + L₁)⁻¹ χ_e — diffusion on the edge space
+cut = edge_local_partition(kc, seed_edge="ver-001", method="hodge_pagerank")
+
+# Hodge heat kernel: e^{-tL₁} χ_e — exponential diffusion on edges
+cut = edge_local_partition(kc, seed_edge="ver-001", method="hodge_heat", t=5.0)
+
+cut.edges         # set of edge IDs in the cluster
+cut.conductance   # edge conductance
+```
+
+The key difference: graph partitioning asks "which vertices are near this vertex?" while edge partitioning asks "which relationships are near this relationship?" — a question that only makes sense in a simplicial complex, not in a plain graph.
+
+## 5. Algebraic topology
+
+Requires `pip install knowledgecomplex[analysis]`.
+
+```python
+from knowledgecomplex.analysis import (
+    boundary_matrices, betti_numbers, euler_characteristic,
+    hodge_laplacian, edge_pagerank, hodge_decomposition, hodge_analysis,
+)
+
+# Boundary matrices (sparse)
+bm = boundary_matrices(kc)
+# bm.B1: (n_vertices × n_edges), bm.B2: (n_edges × n_faces)
+# Invariant: B1 @ B2 = 0 (∂₁ ∘ ∂₂ = 0)
+
+# Betti numbers
+betti = betti_numbers(kc)     # [β₀, β₁, β₂]
+chi = euler_characteristic(kc) # V - E + F = β₀ - β₁ + β₂
+
+# Hodge Laplacian
+L1 = hodge_laplacian(kc)      # B1ᵀB1 + B2B2ᵀ (symmetric PSD)
+# dim(ker L₁) = β₁
+
+# Edge PageRank
+pr = edge_pagerank(kc, "ver-001", beta=0.1)   # (βI + L₁)⁻¹ χ_e
+
+# Hodge decomposition: flow = gradient + curl + harmonic
+decomp = hodge_decomposition(kc, pr)
+# decomp.gradient  — im(B1ᵀ), vertex-driven flow
+# decomp.curl      — im(B2), face-driven circulation
+# decomp.harmonic  — ker(L₁), topological cycles
+
+# Full analysis in one call
+results = hodge_analysis(kc, beta=0.1)
+```
+
+All analysis functions accept an optional `weights` dict mapping element IDs to scalar weights, which factor into the Laplacian as diagonal weight matrices.
+
+## 6. Filtrations
+
+A filtration is a nested sequence of valid subcomplexes: C₀ ⊆ C₁ ⊆ ... ⊆ Cₘ.
+
+```python
+from knowledgecomplex import Filtration
+
+filt = Filtration(kc)
+filt.append({"req-001"})                              # must be valid subcomplex
+filt.append_closure({"ver-001"})                       # auto-closes + unions with previous
+filt.append_closure({"cov-001"})                       # adds face + all boundary
+
+filt.birth("cov-001")   # index where element first appears
+filt.new_at(2)          # elements added at step 2 (Cₚ \ Cₚ₋₁)
+filt[1]                 # set of element IDs at step 1
+
+# Build from a scoring function
+filt2 = Filtration.from_function(kc, lambda eid: some_score(eid))
+```
+
+## 7. Clique inference
+
+Discover higher-order structure hiding in the edge graph:
+
+```python
+from knowledgecomplex import find_cliques, infer_faces
+
+# Pure query — what triangles exist?
+triangles = find_cliques(kc, k=3)
+
+# Fill in all triangles as typed faces
+added = infer_faces(kc, "coverage")
+
+# Preview without modifying
+preview = infer_faces(kc, "coverage", dry_run=True)
+```
+
+## 8. Export and load
+
+```python
+# Export schema + instance to a directory
+kc.export("output/my_complex")
+# Creates: ontology.ttl, shapes.ttl, instance.ttl, queries/*.sparql
+
+# Reconstruct from exported files
+kc2 = KnowledgeComplex.load("output/my_complex")
+kc2.audit().conforms   # True
+```
+
+Multi-format serialization:
+
+```python
+from knowledgecomplex import save_graph, load_graph
+
+save_graph(kc, "data.jsonld", format="json-ld")
+load_graph(kc, "data.ttl")   # additive loading
+```
+
+## 9. Verification and audit
+
+```python
+# Throwing verification
+kc.verify()   # raises ValidationError on failure
+
+# Non-throwing audit
+report = kc.audit()
+report.conforms       # bool
+report.violations     # list[AuditViolation]
+print(report)         # human-readable summary
+
+# Deferred verification for bulk construction
+with kc.deferred_verification():
+    for item in big_dataset:
+        kc.add_vertex(item.id, type=item.type, **item.attrs)
+    # ... add edges, faces ...
+# Single SHACL pass runs on exit
+
+# Static file verification (no Python objects needed)
+from knowledgecomplex import audit_file
+report = audit_file("data/instance.ttl", shapes="data/shapes.ttl",
+                    ontology="data/ontology.ttl")
+```
+
+## 10. Pre-built ontologies
+
+Three ontologies ship with the package:
+
+```python
+from knowledgecomplex.ontologies import operations, brand, research
+
+sb = operations.schema()   # actor, activity, resource
+sb = brand.schema()        # audience, theme
+sb = research.schema()     # paper, concept, note
+```
+
+## Gotchas
+
+| Issue | Detail |
+|---|---|
+| **Slice rule** | Boundary elements must exist before the element that references them. Add vertices → edges → faces. |
+| **Closed triangle** | A face's 3 edges must span exactly 3 vertices in a cycle. An open fan or 4-vertex path will fail. |
+| **`remove_element`** | No post-removal verification. Remove faces before their edges, edges before their vertices. |
+| **Schema after `load()`** | `load()` recovers type names, kinds, attributes, and parent relationships from OWL + SHACL. Full `describe_type()` introspection works after loading. |
+| **Deferred verification** | Inside the context manager, intermediate states need not be valid. Verification runs once on exit. |
+| **Face orientation** | Boundary matrix signs are computed internally to guarantee ∂₁∘∂₂ = 0. The orientation is consistent but not guaranteed to match external conventions. |