Geometry-Aware Simplicial Message Passing

Topological neural networks have emerged as effective tools for modeling higher-order relational structures beyond pairwise graphs, including hypergraphs, simplicial complexes, and cell complexes. However, existing Weisfeiler-Leman type expressivity analyses are typically developed on different structural domains and rely on domain-specific neighborhood systems, making their expressive powers difficult to compare within a common formalism. In this paper, we introduce the Combinatorial Complex Weisfeiler-Leman (CCWL) framework, a unified expressive power refinement defined on combinatorial complexes. By exploiting the ability of combinatorial complexes to represent both set-type relations and part-whole hierarchies, CCWL performs topological color refinement through four structural neighborhoods: boundary, co-boundary, lower adjacency, and upper adjacency. We show that, under specified lifting maps, CCWL can simulate several domain-specific WL-type refinements, thereby providing a common theoretical baseline for analyzing topological message passing. We further study the neighborhood sufficiency problem and prove that, under explicit coverage conditions, a reduced refinement using only lower- and upper-adjacent bridge information preserves the distinguishing power of the full four-neighborhood CCWL refinement. Guided by this theoretical result, we instantiate the reduced refinement as the Combinatorial Complex Isomorphism Network (CCIN). Experiments on synthetic and real-world benchmarks demonstrate that CCIN achieves competitive performance against representative graph and topological neural network baselines. Ablation studies and resource-efficiency analyses further support the effectiveness of the proposed lower/upper-neighborhood design.

Whitney's theorem allows isomorphism testing for connected simple graphs, apart from $K_3$ and $K_{1,3}$, to be formulated as distinguishing their line graphs. However, the relation between fixed-dimensional Weisfeiler--Leman (WL) expressivity on line graphs and on their roots remains unresolved. We study this relation through Implicit Line-Graph WL (ILG-$k$-WL), which is exactly $k$-WL on $L(G)$, executed over the edges of $G$ with line-graph relations derived from endpoint incidence and without explicitly constructing $L(G)$. On the Whitney-general class, the relation between root-domain and line-graph WL depends on $k$. For $k=1,2$, ILG-$k$-WL adds no distinguishing power beyond root-domain $1$-WL and misses some pairs that $1$-WL separates. For $k=3$, we prove the backward containment $L(G)\equiv_{3\text{-WL}}L(H)\Rightarrow G\equiv_{3\text{-WL}}H$. Strongly regular witness pairs, including the Shrikhande/rook pair, show that ILG-$3$-WL is strictly more expressive than $3$-WL. The backward containment also extends to disconnected graphs with no isolated vertices when every connected component is Whitney-general. Deterministic ILG-$3$-WL separates all three substructure-counting witness pairs, all $105$ pairs in SR25, and $359$ of $400$ BREC pairs. An untrained dense ILG-$3$-GNN gives the same pairwise verdicts on these evaluations.

Graphs with a simple spectrum admit cubic-time isomorphism testing, yet we prove that for every natural number $k$, the $k$-Weisfeiler-Leman ($k$-WL) test cannot distinguish all non-isomorphic graphs with a simple spectrum. As the WL hierarchy upper-bounds the distinguishing power of widely-used Graph Neural Networks (GNNs), this incompleteness applies to all such GNNs, ruling out completeness for every $k$-WL-aligned GNN family. To close this gap, we introduce PRiSM (Partition, Refine, Solve, Match), the first provably complete canonicalization of simple-spectrum eigendecompositions. PRiSM obtains the completeness guarantee that prior canonicalizations provably lack, and resolves the open problem of achieving complete expressivity on simple-spectrum graphs. When composed with DeepSets or a Transformer, PRiSM achieves universal approximation on simple-spectrum graphs, justifying the use of canonicalized Laplacian positional encodings. Empirically, PRiSM performs comparably to or outperforms existing spectral canonicalizations on graph regression, classification, and expressivity