ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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An irrational GIN self-weight separates two bounded color neighborhoods

Example

Let the countable feature set be $X = {r, b}$ with code $ϕ (r) = 1$ , $ϕ (b) = 2$ , the base- $2$ code $ϕ (x) = 2^{ν (x)}$ for $ν (r) = 0, ν (b) = 1$ . Set the GIN self-weight parameter to $ε = \sqrt{2}$ . On a center feature $c$ and a one-neighbor multiset $S$ , the scalar preactivation is $J_{ε} (c, S) = (1 + ε) ϕ (c) + \sum_{x \in S} ϕ (x)$ . For the distinct center/neighbor pairs $(r, {⁣ {b} ⁣})$ and $(b, {⁣ {r} ⁣})$ , the preactivations are $3 + \sqrt{2}$ and $3 + 2 \sqrt{2}$ , respectively, so this GIN sum code separates the pair under the size bound $∣ S ∣ \leq 1$ .

Example references

finite multisets and their sizes

the GIN sum preactivation

the countable bounded-multiset encoding theorem

Xu, Hu, Leskovec, and Jegelka, “How Powerful are Graph Neural Networks?”, §4.1 and Corollary 6

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

Facts & Assumptions

Given: The two feature values, code, and center/neighbor pairs in the Example. Verify the irrationality condition and the two preactivation values.

[F1]

Finite multisets as finitely supported multiplicity functions: A multiset records each feature with its number of occurrences.

[F2]

Graph Isomorphism Network update rule: A GIN preactivation is $(1 + ε) h_{v} + \sum_{u \in N (v)} h_{u}$ before the layer MLP.

[F3]

Countable bounded multisets admit injective sum encodings: For countable features and bounded neighbor-multiset size, an irrational self-weight makes the coded center/multiset sum injective.

Verification

1.1givenalgebracontradiction

The number $\sqrt{2}$ is irrational: if $\sqrt{2} = p / q$ for coprime positive integers $p, q$ , then $p^{2} = 2 q^{2}$ makes $p$ even, say $p = 2 s$ , and substitution gives $q^{2} = 2 s^{2}$ , so $q$ is even as well, contradicting coprimality.

2.1givenF1F2step 1.1algebra

By [F2], $J_{\sqrt{2}} (r, {⁣ {b} ⁣}) = (1 + \sqrt{2}) \cdot 1 + 2 = 3 + \sqrt{2}$ , while $J_{\sqrt{2}} (b, {⁣ {r} ⁣}) = (1 + \sqrt{2}) \cdot 2 + 1 = 3 + 2 \sqrt{2}$ . Their difference is $\sqrt{2} \neq 0$ , so the distinct pairs have distinct preactivations.

3.1givenF1F3step 1.1step 2.1algebra∎

The feature set has two elements and both neighbor multisets have size one. Thus the example meets the countability and bound hypotheses of [F3], with irrational $ε$ established in step 1.1; the computed separation is a concrete instance of that injective coding condition.

Depends on

Used by

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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)