2-FWL Test: Enhanced Graph Isomorphism

• The paper introduces 2-FWL as a higher-order graph isomorphism test that leverages pair-based aggregation to simulate all 2-length walks and capture complex substructures.
• 2-FWL surpasses classical 1-WL tests by distinguishing non-isomorphic graphs through detailed 3-node interactions and enhanced counting of motifs.
• Efficient variants like Neighborhood²-FWL and Co-Sparsify reduce the cubic time complexity, enabling scalable, maximally expressive GNN architectures on large graphs.

The 2-FWL test, or 2-dimensional Folklore Weisfeiler–Lehman test, is a higher-order color refinement procedure and a powerful graph isomorphism test. It serves as both a theoretical upper bound for message-passing GNN expressivity and a core primitive in the design of higher-order graph neural networks. By operating on pairs of nodes and aggregating information about paths and substructures, 2-FWL significantly surpasses the classical 1-WL (node-color refinement) in its ability to distinguish non-isomorphic graphs and to encode higher-order combinatorial patterns. The 2-FWL test is equivalent in expressive power to the 3-WL test. Recent research focuses on making 2-FWL tractable for large graphs, enabling scalable, maximally expressive GNN architectures.

1. Formal Definition and Update Mechanism

The 2-FWL test maintains a coloring $\mathcal{C}^{(l)}(u,v)$ of all ordered pairs $(u,v)$ in a graph $G=(V,E)$. Initialization encodes node and edge attributes:

$\mathcal{C}^{(0)}(u, v) = (x(u), x(v), e(u,v))$

where $x(\cdot)$ are node features and $e(u,v)$ is the edge feature.

The iterative update at each layer $l \ge 1$ aggregates over all potential intermediate nodes $t$:

$$\mathcal{M}^{(l)}(u, v) = \big\{\big(\mathcal{C}^{(l-1)}(u, t),\, \mathcal{C}^{(l-1)}(t, v)\big)\ \big|\ t \in V \big\}$$

$$\mathcal{C}^{(l)}(u, v) = \mathrm{Hash}\Big(\mathcal{C}^{(l-1)}(u, v),\, \mathrm{SortMultiset}(\mathcal{M}^{(l)}(u, v))\Big)$$

where $\mathrm{Hash}$ is an injective operation, which in practice can be implemented with MLP + sum aggregation. This amounts to simulating all walks of length 2 between node pairs, capturing not only edge-level but also path-level and motif-level structure. The update is repeated until the coloring stabilizes.

The time complexity per layer is $\Theta(n^3)$ due to $n^2$ pairs each aggregating over $n$ intermediates (Chen et al., 16 Nov 2025, Feng et al., 2023, Puny et al., 2020, Zhang et al., 2023).

2. Expressive Power and Theoretical Position

2-FWL’s ability to distinguish non-isomorphic graphs exceeds the 1-WL test; equivalently, $k$-FWL is as expressive as $(k+1)$-WL, so $2$-FWL matches $3$-WL. This enables 2-FWL to:

• Count triangles and $k$-cycles (for $3 \leq k \leq 6$), 4-paths, and richer motifs.
• Distinguish graphs (including strongly regular or Fürer graphs) indistinguishable by 1-WL or even subgraph Weisfeiler–Lehman (SWL) techniques (Feng et al., 2023, Zhang et al., 2023).

There is a provable gap between the distinguishing power of node-based SWL and 2-FWL: even the most expressive node-subgraph-based SWLs, such as SSWL, fall strictly short of 2-FWL (and hence 3-WL) (Zhang et al., 2023). This expressivity comes at cubic computational cost, motivating research into efficient yet fully expressive sparsifications.

3. Computational Complexity and Efficient Variants

The standard 2-FWL test is prohibitively expensive for large graphs due to its $O(n^3)$ runtime and $O(n^2)$ space requirements. Approaches to mitigation include:

• Local windowing: Neighborhood$^2$-FWL (N$^2$-FWL) restricts aggregation to neighborhood pairs, reducing time to $O(n^2 d^2)$ when the average node degree $d$ is small. N$^2$-FWL achieves practical implementations matching or exceeding the expressiveness of full 2-FWL in many settings (Feng et al., 2023).
• Connectivity-guided sparsification (Co-Sparsify): Leverages graph structure by restricting 3-node interactions to biconnected components and limiting message passing to within connected components. This yields sub-cubic or near-quadratic cost for graphs with small biconnected components, without loss of 2-FWL distinguishing power (Chen et al., 16 Nov 2025).

The following table summarizes per-iteration complexities for major 2-FWL variants:

Variant	Time per Iteration	Space
Standard 2-FWL	$O(n^3)$	$O(n^2)$
Neighborhood$^2$-FWL	$O(n^2 d^2)$ (sparse)	$O(n^2)$
Co-Sparsify	$O\big(\sum_i n_i^2+\sum_j b_j^3\big)$	$O(n^2)$

Here, $n_i$ is the size of connected component $i$, $b_j$ is the size of biconnected block $j$.

4. Graph Connectivity and the Role of Biconnected Components

2-FWL leverages both 2-node and 3-node interactions:

• 2-node interactions (cases $t=u$ or $t=v$) capture connectivity and reachability between nodes.
• 3-node interactions, for $u$, $t$, $v$ all distinct, are essential for detecting two or more vertex-disjoint $u$–$v$ paths. This is necessary for detecting cycles and the property of biconnectivity.

A subgraph is biconnected if the removal of any single node does not disconnect it, i.e., for all $u\neq v$ in the block, the vertex-connectivity $\kappa_G(u,v)\geq2$. Within biconnected components, 3-node interactions are required to fully capture structural nuances; outside those components, 2-node message passing suffices (Chen et al., 16 Nov 2025).

This structural insight underpins Co-Sparsify: 3-node message passing is pruned outside blocks, with theoretical guarantees of preserved expressivity.

5. Algorithmic Alignments and GNN Implementations

The update mechanism of 2-FWL has direct algorithmic correspondence with higher-order GNN architectures:

• Pair-based GNNs such as Pairwise Permutation-invariant GNN (PPGN) and related HOGNNs implement 2-FWL color propagation as neural message-passing over all node pairs.
• Low-Rank Global Attention (LRGA), when combined with random encoding (RGNN), is shown theoretically to simulate a full 2-FWL iteration through a composition of monomial MLPs and low-rank matrix attention modules (Puny et al., 2020).
• N$^2$-GNN realizes the N$^2$-FWL paradigm, achieving near 3-WL power with $O(n^2)$ space (Feng et al., 2023).

Sparsification schemes that restrict message passing as in Co-Sparsify preserve full 2-FWL expressivity if aggregation and update functions remain injective and initialization is consistent (Chen et al., 16 Nov 2025).

6. Empirical Performance and Benchmarking

Empirical studies confirm the theoretical advantages of efficient, fully expressive 2-FWL-based architectures:

• On synthetic substructure counting (tailed triangles, chordal cycles, etc.), sparsified PPGN variants match or exceed standard pair-GNNs, attaining state-of-the-art normalized mean absolute error (Chen et al., 16 Nov 2025).
• On ZINC and QM9 regression benchmarks, CoSp-PPGN+RRWP variants produce lower MAE than unsparsified models.
• Efficiency gains are substantial: per-epoch runtime and GPU memory usage drop by 13–60% and 12–52%, respectively, with near-zero preprocessing overhead (Chen et al., 16 Nov 2025).
• N$^2$-GNN achieves state-of-the-art on ZINC-Subset and BREC benchmarks with $O(n^2)$ memory (Feng et al., 2023); LRGA-augmented architectures also achieve new state-of-the-art results on diverse tasks (Puny et al., 2020).

7. Implications and Limitations

The 2-FWL test formalizes the upper limit of expressiveness for subgraph GNNs operating at $O(n^2)$ space. While node-subgraph GNNs (such as SSWL or PSWL) can approximate 2-FWL power locally, they are provably weaker in distinguishing certain hard graph pairs (Zhang et al., 2023).

Practical adoption of 2-FWL-inspired GNN architectures must consider scalability constraints, as underlying cubic time remains prohibitive in the worst case. Local approximation schemes (e.g., N$^2$-FWL, Co-Sparsify), careful data structure design, and injective implementation of aggregation functions are necessary for scalability and full theoretical guarantees. Remaining bottlenecks include optimization instability at scale and potential capacity issues in over-parameterized neural approximations.

A plausible implication is that closing the remaining efficiency–expressivity gap for massive graphs may require further innovations, such as adaptive sampling, randomized sketching, or fundamentally new higher-order GNN primitives.