Uniqueness of tuple grouping in 3-WL for 3D graphs

Establish whether, in the 3-dimensional Weisfeiler–Lehman (3-WL) update that aggregates colors from coordinate-replacement neighbors, the grouping of the three neighbor tuples {Φ1(v,j), Φ2(v,j), Φ3(v,j)} into an associated information set for the same external node j is uniquely determined.

Background

The paper contrasts 3-FWL and 3-WL. In 3-FWL, the update directly collects the triplet of neighbor tuple colors corresponding to the same external node j, avoiding ambiguity. In 3-WL, the update collects three separate multisets of neighbor tuple colors, one per coordinate replacement, which introduces a potential grouping ambiguity.

Resolving whether this grouping is unique is critical to understanding if 3-WL can avoid the exchange/turn-over uncertainties identified for graph generation and thus whether it can reliably recover 3D structure.

References

By comparing with function \ref{con: 3-FWL}, we can find that the biggest difference between them is that the update function of 3-WL cannot directly obtain which \Phi_i(v,j)s are the associated information belonging to the same node. In other words, it is difficult to find {\Phi_1(v,j),\Phi_2(v,j),\Phi_3(v,j)}\in inf(v, j). Moreover, it is unknown whether this grouping is unique.

— Is 3-(F)WL Enough to Distinguish All 3D Graphs? (2402.08429 - Xu, 24 Jan 2024) in Section “Does 3-WL have tricks?”, paragraph discussing differences between 3-WL and 3-FWL