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One-dimensional Weisfeiler–Leman refinement is invariant under graph isomorphism

Statement

Let $G$ and $H$ be finite vertex-labeled simple undirected graphs with labels in a common set, and let $φ : G \to H$ be a label-preserving graph isomorphism. At every finite round $t$ , corresponding vertices have equal one-dimensional Weisfeiler–Leman colors: $c_{t}^{G} (v) = c_{t}^{H} (φ (v))$ for every $v \in V (G)$ . Consequently their graph color multisets agree at every round, so the refinement does not distinguish isomorphic graphs.

Statement references

finite vertex-labeled graphs and neighborhoods

label-preserving graph isomorphisms

multiset equality and multiplicity

one-dimensional color refinement

Morris et al., “Weisfeiler and Leman Go Neural,” §2

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

Li and Leskovec, Graph Representation Learning, Chapter 5 §5.3.3

Facts & Assumptions

Given: The finite labeled graphs $G, H$ and label-preserving isomorphism $φ : G \to H$ in the Statement. Prove the color and histogram claims.

[F1]

Label-preserving graph isomorphisms: $φ$ is a bijection preserving labels and adjacency in both directions.

[F2]

Finite-round one-dimensional Weisfeiler–Leman color refinement: The round- $t + 1$ color is the ordered pair of the vertex's round- $t$ color and the multiset of round- $t$ colors of its neighbors.

[F3]

Finite multisets as finitely supported multiplicity functions: Equality of finite multisets is equality of multiplicities at every element.

Proof

1.1givenF1F2inductionalgebra

By induction on $t$ , $c_{t}^{G} (v) = c_{t}^{H} (φ (v))$ for every $v \in V (G)$ . At $t = 0$ this is label preservation from [F1]. If it holds at $t$ , bijectivity and adjacency preservation in [F1] give a bijection $N_{G} (v) \to N_{H} (φ (v))$ , and corresponding neighbors have equal round- $t$ colors by the induction hypothesis. Thus the neighbor-color multisets agree; the center colors agree as well, so the ordered-pair update [F2] gives equality at round $t + 1$ .

2.1givenF1F3step 1.1algebra∎

The same vertex bijection therefore sends every occurrence of each round- $t$ color in $G$ to an occurrence of that color in $H$ , and its inverse gives the reverse correspondence. Hence the two graph color multisets have equal multiplicities for each color by [F3]. Their round- $t$ histograms agree for every finite $t$ , so no round distinguishes them.

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Sources

Christopher Morris et al., Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks (standard reference, not scraped)
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)