This document discusses the Weisfeiler-Lehman graph isomorphism test and its generalization k-WL. It explains that k-WL colors k-tuples of vertices and can distinguish some graphs that (k-1)-WL cannot. While k-WL and GNNs are related, their relationship is complicated by GNNs using continuous values. The document also covers runtime, applications to regular graphs and minors, and relationships to graph properties like planarity.

1. Weisfeiler-Lehman by the Numbers

2. Weisfeiler-Lehman by the Numbers
1)

3. Weisfeiler-Lehman by the Numbers
2)

4. Weisfeiler-Lehman by the Numbers
3)

5. Weisfeiler-Lehman by the Numbers
4)

6. Weisfeiler-Lehman by the Numbers
5)

7. Weisfeiler-Lehman by the Numbers
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8. Weisfeiler-Lehman by the Numbers
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9. Weisfeiler-Lehman by the Numbers
8) Continue until there is no change in the labels (up to bijection).
9) “If there exists c such that the two graphs have a different number
of nodes coloured c, then the two graphs are non-isomorphic.”
• You actually don’t have to wait until termination to check this.
• This is not an isomorphism test – it is a non-isomorphism test.
• Can’t we just negate the result of the test at termination?

10. No
• WL would then say that these are isomorphic:
• Alright, but maybe it works on connected graphs…

11. No
• WL would then say that these are isomorphic:
• So maybe we do need a better test.
• Still, this is remarkably effective for such a simple algorithm: all n nodes of
a random n-node graph receive different colours after just 2 iterations of
WL with probability 1-
7 1
𝑛
(Babai et al., 1980).

12. k-WL
• Generalization of WL in which k-tuples of vertices are coloured.
• Base case: Ԧ
𝑣 and 𝑤 receive same initial colour iff vi↔wi yields a subgraph
isomorphism.
• There are families of graphs Gk and Hk such that Gk and Hk are indistinguishable
by k-WL, but distinguishable by (k+1)-WL.
• Be careful: there are people who consider k-WL to be a generalization in which
the colours are k-dimensional. But there are known equivalences between the
two formulations.
• This would seem to establish a natural connection between k-WLs and GNNs in k-
dimensions.
• Thm (Morris et al., 2019): GNNs’ expressive power on graph isomorphism task is
bounded above by classical WL regardless of dimensionality or network depth,
but relatively simple architectures can attain this bound.
• Very tricky to correlate the two, because GNNs use putatively continuous-valued
labels, a generalization of WL that has also been explored (Orsini et al., 2015).

13. More fun with k-WL
• Run-time complexity of k-WL is O(nk+1 log n).
• Prob. that 2-WL identifies two d-regular graphs of n nodes → 1 as
n →∞ (Bollobas, 1982).
• 2-WL cannot distinguish any two strongly regular graphs of the same
parameters (Grohe and Neuen, 2021).
• WL-dimension: least k such that G is distinguishable by k-WL from all
graphs not isomorphic to it.
• Trivial: every n-node graph has WL-dimension ≤ n-1.

14. 1
4
Graph Deletion & Contraction
Important fact: κ(G) = κ(G-{e}) + κ(G{e})

15. More fun with k-WL
• For all h-node graphs H, h > 0, there is a k > 0 such that every graph
that excludes H as a minor has WL-dimension ≤ k (Grohe, 2012).
• All planar graphs exclude K5 (Kuratowski, 1930).
• Therefore all planar graphs have bounded WL-dimension.