degree concept

Definition	Of a node in a graph, the number of edges it has. Chapter 0 section 0.10.
Example	A node with edges to three others has degree 3, and the degrees of a graph sum to twice its edge count.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `degree`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.19.
Assumptions and scope	The number of edges a node has. Summed over the graph the degrees count every edge twice, the sum is even, and no node has degree more than the number of other nodes. The radius of a spectral embedding tracks the degree at correlation 0.92 to 0.99 on the substrates tested, which is the measured half of chapter 3’s claim that the magnitude is degree noise and the angle is what the geodesic-rank reader reads.
Prior art	none recorded
Evidence	`lean/DataMiningAsObservation/Degree.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The number of edges at a node.

Equation

Book equation 0.19.

\[L=I-D^{-1/2}AD^{-1/2},\qquad D=\operatorname{diag}(d_1,\dots,d_n),\qquad L\,u_k=\lambda_k u_k,\ \ 0=\lambda_1\le\lambda_2\le\cdots.\]

Book equation 3.3.

\[X_i=r_i\,\theta_i,\qquad r_i\ \to\ \frac1{\sqrt{d_i}},\qquad \theta_i=\frac{X_i}{\|X_i\|}\in S^{m-1}.\]

Conditions

The number of edges a node has. Summed over the graph the degrees count every edge twice, the sum is even, and no node has degree more than the number of other nodes.
The radius of a spectral embedding tracks the degree at correlation 0.92 to 0.99 on the substrates tested, which is the measured half of chapter 3’s claim that the magnitude is degree noise and the angle is what the geodesic-rank reader reads.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Chapter 0 section 0.10 and chapter 3 section 3.3 of Data Mining as Observation, with the radius-against-degree measurement in the-angular-observer, the-angular-observer/README.md:135-139.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 3 section 3.3	radius against degree correlation 0.92 to 0.99	`the-angular-observer/README.md:135-139`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Degree.lean, theorems sum_degrees, degree_lt_card, sum_degrees_even, average_degree, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer L, chapters 0, 1, 3, 4, 8, 9.

Laplacian; geodesic distance; hubness; recognizer.

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.

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