AlgorithmAlgorithm: 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.

Finite-depth message-passing forward pass

Definition

Given a finite vertex-labeled graph $G = (V, E, ℓ)$ , a finite nonnegative integer $K$ , feature spaces and maps of the types in the aggregation-based message-passing definition, compute a graph output as follows. Initialize $h_{v}^{0} = η (ℓ (v))$ for every $v \in V$ . For each $k = 1, \dots, K$ , first compute $a_{v}^{k} = A_{k} ({⁣ {h_{u}^{k - 1} : u \in N_{G} (v)} ⁣})$ for every $v \in V$ , then set $h_{v}^{k} = U_{k} (h_{v}^{k - 1}, a_{v}^{k})$ for every $v \in V$ . Return $R ({⁣ {h_{v}^{K} : v \in V} ⁣})$ . When $K = 0$ , return $R ({⁣ {η (ℓ (v)) : v \in V} ⁣})$ without a layer update.

Definition references

finite labeled graphs and neighborhoods

the typed aggregation, update, and readout maps

Xu, Hu, Leskovec, and Jegelka, “How Powerful are Graph Neural Networks?”, §2

Morris et al., “Weisfeiler and Leman Go Neural,” §3

Li and Leskovec, Graph Representation Learning, Chapter 5 §5.3.2

Depends on

Used by

Nothing in the library uses this result yet.

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)
Christopher Morris et al., Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks (standard reference, not scraped)
Pan Li and Jure Leskovec, Graph Representation Learning (standard reference, not scraped)