DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Graph Isomorphism Network update rule

Definition

Fix a finite depth $K \in N_{0}$ , positive integer dimensions $d_{0}, \dots, d_{K}$ , an initial embedding $η : Σ \to R^{d_{0}}$ , scalars $ε_{1}, \dots, ε_{K} \in R$ , and maps $M_{k} : R^{d_{k - 1}} \to R^{d_{k}}$ for $1 \leq k \leq K$ . Each $M_{k}$ is a multilayer perceptron, meaning a finite composition of affine maps between finite-dimensional real vector spaces and coordinatewise applications of a fixed scalar activation. On a finite vertex-labeled simple undirected graph $G$ , initialize $h_{v}^{0} = η (ℓ (v))$ and update

h_{v}^{k} = M_{k} ⁣ ((1 + ε_{k}) h_{v}^{k - 1} + \sum_{u \in N_{G} (v)} h_{u}^{k - 1}) (1 \leq k \leq K) .

For a chosen output set $Y$ , the graph readout is a map $R : M_{f i n} (R^{d_{K}}) \to Y$ on the multiset of final vertex features. The neighbor sum counts each neighbor vertex once, so equal feature vectors at different neighbors contribute with their multiplicity. This definition imposes no injectivity condition on the perceptrons or the readout and makes no claim that a fixed finite architecture realizes an arbitrary injection.

Definition references

aggregation-based updates and multiset graph readouts

Xu, Hu, Leskovec, and Jegelka, “How Powerful are Graph Neural Networks?”, §4.1, Eq. (4.1)

Li and Leskovec, Graph Representation Learning, Chapter 5 §5.3.4

Morris et al., “Weisfeiler and Leman Go Neural,” §3, Eq. (5)

Depends on

Aggregation-based message-passing graph neural networks

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka, How Powerful are Graph Neural Networks? (standard reference, not scraped)
Pan Li and Jure Leskovec, Graph Representation Learning (standard reference, not scraped)
Christopher Morris et al., Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks (standard reference, not scraped)