Keep the mechanism close.

Notation, evidence and the boundary around the toy.

A compact glossary

Node / edge

An entity / a chosen relation between entities. Our edges are undirected.

Feature x / hidden state h

The observed input / the intermediate representation. Each has one dimension in this toy.

Neighbor set N(i) / degree dᵢ

The nodes adjacent to i / their count. Degree counts present edges, not gate strength.

Message / aggregate / update

A transformed neighbor contribution / their order-independent combination / the new receiver state.

Layer ℓ / parameters θ

A stage of computation / coefficients fitted during training in a learned model. Our coefficients are stipulated.

Receptive field

The input region a state can depend on. For our local synchronous model at L layers, it is bounded by L hops.

Equivariant / invariant

Outputs reorder with inputs / an output stays unchanged under that reordering.

Gate eᵢⱼ

A synthetic multiplier on a message. Neither an electrical parameter nor a normalized attention score.

Primary sources

Inspected on 4 September 2026. Source claims are identified by CL IDs. Toy calculations are derived from declared assumptions; these papers do not validate our grid-shaped example.

1. Gilmer et al. · Neural Message Passing for Quantum Chemistry (2017)

   §2, equations 1–3: message/update functions, attributed graphs and readout. CL-001,002,015,016.

2. Battaglia et al. · Relational inductive biases, deep learning, and graph networks (2018)

   §3.2: graph representation, shared functions and invariant aggregation. §4.3/Figure 7: locality without global channels. CL-003,004.

3. Kipf & Welling · Semi-Supervised Classification with Graph Convolutional Networks (2017)

   §2, equation 2: a specific symmetric-normalized GCN. This is not our implemented scalar mean. CL-005.

4. Li, Han & Wu · Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning (2018)

   “Laplacian Smoothing” and “When GCNs Fail,” Figure 2/Theorem 1. Scope matters. CL-006.

5. Oono & Suzuki · Graph Neural Networks Exponentially Lose Expressive Power for Node Classification (2020)

   §3.3 Corollary 2; §4 Theorem 2: architectural and spectral/weight conditions. CL-006.

Reproduce the computation

The canonical model module (local copy) is shared by the page, the static figures and the command-line lab. Read its numerical checks (local copy). In the project folder, run:

node code-lab/run.mjs
node code-lab/run.mjs --check
cd site
npm run build
npm test

The lab records deterministic outputs in code-lab/outputs.json. It uses JavaScript numbers, no random source, no dataset, no network and no training. Full-precision calculations drive the display; tables round to three decimals. The longer averaging run repeats the same fixed operator and is not a sequence of training updates.

Build-time KaTeX supplies local HTML and MathML; no math CDN is needed. The complete reading edition includes all explanations, comparisons and worked answers with zero scripts.