neighbourhood graph concept
| Definition | A graph on a dataset that joins each row to its k nearest rows, or to every row within a radius. Chapter 0 section 0.10. Also k-nearest. |
|---|---|
| Example | A 5-nearest-neighbour graph on 1000 rows has 5000 out-edges, five from each row. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id neighbourhood-graph, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.20. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Graph.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.20.
\[\Psi_i=\left(\frac{u_k(i)}{\sqrt{\lambda_k\,d_i}}\right)_{k\ge2},\qquad \|\Psi_i-\Psi_j\|^{2}=R(i,j)=\frac{C(i,j)}{\operatorname{vol}(G)}.\]
Conditions
- A graph on a dataset that joins each row to its k nearest rows, or to every row within a radius. Every node of a k-nearest graph has out-degree k, the mutual neighbours of a node are at most k, and being mutual neighbours is symmetric.
- As the graph grows on a manifold its low Laplacian eigenvectors converge to the smoothest functions on the manifold, so the embedding becomes a picture of the manifold, and the hubs of the graph are a property of the queries that built it.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.10 of Data Mining as Observation, with
the substrate table in
the-angular-observer/README.md:99-109.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Graph.lean,
theorems outDegree, mutual_symm,
mutual_sub, mutualNbrs_card_le,
mem_mutualNbrs, at observation-data-mining f3914f0; what
the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation chapters 0, 3, 9, 11.
Related
graph; degree; Laplacian; spectral embedding; geodesic distance; hub.
See also
Book equations stated beside the entry’s terms, not defining it: 0.19, 3.3.
Ledger rows that cite the entry’s records without naming it: NEG-1.
Sources-table rows that share a record with the entry without naming it: chapter 3 section 3.3, chapter 9 section 9.1, chapter 9 section 9.2.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.