Understanding Truncated Positional Encodings for Graph Neural Networks

Amir Nayyeri; James Flora; Mitchell Black; Weng-Keen Wong

arxiv: 2606.13671 · v1 · pith:DN5OT2HGnew · submitted 2026-06-11 · 💻 cs.LG

Understanding Truncated Positional Encodings for Graph Neural Networks

James Flora , Mitchell Black , Weng-Keen Wong , Amir Nayyeri This is my paper

Pith reviewed 2026-06-27 07:07 UTC · model grok-4.3

classification 💻 cs.LG

keywords positional encodingsgraph neural networksexpressive powertruncationspectral encodingswalk-based encodingsWeisfeiler-Lehman testgraph distinction

0 comments

The pith

Truncating positional encodings for graph neural networks makes different families unequal in expressive power.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the known equivalence between spectral and walk-based positional encodings holds only for their full, computationally expensive versions. Once truncated to the first k components, as done in practice, these families separate in what graphs a GNN can distinguish. Truncated spectral encodings drop to the level of the basic 1-WL test, while walk-based versions keep more power. Even closely related spectral choices, such as k-harmonic distances, prove distinct from each other after truncation. Experiments confirm that combining several truncated families outperforms any single one on real datasets.

Core claim

Under uniform truncation to the first k components, spectral positional encodings (Laplacian eigenspaces, effective resistance) are no longer stronger than the 1-WL test, while walk-based encodings (adjacency powers) retain higher power, breaking the equivalence that exists for complete versions. Different truncated spectral families, including k-harmonic distances, also differ from one another in expressive power.

What carries the argument

Uniform truncation of positional encoding vectors to the first k entries, compared by how much they increase a GNN's ability to distinguish non-isomorphic graphs.

If this is right

Truncated spectral positional encodings match the distinction power of the 1-WL test.
Walk-based truncated encodings remain strictly stronger than the 1-WL test.
Even within the spectral family, different truncation choices such as k-harmonic distances produce distinct expressive powers.
Combining truncated encodings from multiple families improves empirical performance over any single family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The choice of which family to truncate may matter more than the choice of k itself.
Theoretical guarantees written for complete encodings cannot be assumed to carry over to the versions actually used in deployed models.
Designers may need family-specific rules for choosing k rather than a single shared value.

Load-bearing premise

The analysis assumes the same truncation parameter k applies to every family and that the GNN processes the resulting vectors in comparable ways when distinguishing graphs.

What would settle it

Two graphs that the 1-WL test cannot separate, yet a GNN equipped only with the first k Laplacian eigenvectors as positional encodings can separate.

Figures

Figures reproduced from arXiv: 2606.13671 by Amir Nayyeri, James Flora, Mitchell Black, Weng-Keen Wong.

**Figure 1.** Figure 1: Top row: The four smallest eigenvectors of the graph P4 × K10. Bottom row: The four smallest eigenvectors of the graph P4 × K5,5. Comparing the two rows shows there is a bijection between the nodes of P4 × K10 and P4 × K5,5 that preserves the values of the eigenvectors. These two graphs are indistinguishable by 4-EP-WL. Theorem 3.1 ((Xu et al., 2018; Morris et al., 2019)). Two graphs are indistinguishable … view at source ↗

**Figure 2.** Figure 2: The graphs G (the length 9 cycle graph) and H (3 copies of the length 3 cycle graph) are indistinguishable by the WL test, but distinguishable by any Sparse-k-Harmonic-WL test G with connected components G1, . . . , Gk, L k+ G =    L k+ G1 0 . . . 0 L k+ G2 . . . . . .    This follows from the fact that the eigenvectors of a disconnected graph are the eigenvectors of each of its connected components … view at source ↗

**Figure 3.** Figure 3: Two non-isomorphic trees that Sparse-Biharmonic-WL can distinguish in 1 iteration but Sparse-Resistance-WL cannot distinguish in o(n) iterations. Proof of Theorem 4.7. Let n be any positive integer that is divisible by 12. We consider the pair of rooted trees G and H where G is a root connected to three paths of length n/3 and H is a root connected to two paths of length n/4 and one path of length n/2; see… view at source ↗

**Figure 4.** Figure 4: Two non-isomorphic trees G and H with n vertices that Sparse-Biharmonic-WL takes o(n) iterations to distinguish. Let T be the complete tree consisting of n nodes and P be a path of n nodes. To prove this, we will use a variant of Lemma D.8 for the sparse ψ WL test. For an edge positional encoding ψ and two graphs G and H, we define a ψ-preserving isomorphism as an isomorphism σ : VG → VH such that for each… view at source ↗

read the original abstract

Positional encodings (PEs) enhance the power of graph neural networks (GNNs), both theoretically and empirically. Two of the most popular families of PEs - spectral (e.g., Laplacian eigenspaces, effective resistance) and walk-based (polynomials of the adjacency matrix) - are theoretically equivalent in expressive power, with expressivity between the 1-WL and 3-WL tests. However, this equivalence assumes the GNN uses the "complete" version of these PEs, which requires $O(n^3)$ time and space complexity. Instead, practitioners commonly use truncated variants of these encodings, such as the first $k$ eigenspaces or powers of the adjacency matrix. However, the theoretical properties of these truncated PEs are unknown. In this work, we initiate the study of these truncated PEs. Theoretically, we show that, under truncation, several families of PEs are fundamentally different in expressive power. As a corollary, we show that truncated spectral PEs are no longer stronger than the 1-WL test. We also study a family of spectral PEs, the $k$-harmonic distances, to highlight the differences in expressive power of even closely related truncated PEs. Finally, we experimentally show that a mix of truncated PEs is preferable to any single family on real-world datasets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Truncation breaks the known equivalence between spectral and walk-based PEs, but the comparison hinges on whether the same k really gives equivalent inputs to the GNN.

read the letter

The main point here is that complete spectral and walk-based positional encodings are equivalent in power (between 1-WL and 3-WL), but their truncated versions are not. The paper shows truncated spectral encodings drop to no stronger than 1-WL, and that even closely related truncated families like k-harmonic distances differ from the others. That distinction is new; prior work focused on the full versions.

The theoretical part is the real contribution. They initiate the comparison under truncation and give a corollary that directly affects how people use Laplacian eigenvectors in practice. The experimental claim that mixing families works better on real datasets is a useful practical takeaway, though the abstract gives no setup details.

The soft spot is the truncation parameter itself. The stress-test note is on target: keeping the k smallest eigenvectors is not obviously the same information budget as keeping the first k powers of the adjacency matrix when both are fed to a standard message-passing GNN. If the paper does not show that the effective dimension or the WL test application is matched across families, the claimed difference could partly reflect unequal truncation strength rather than an intrinsic property. The abstract does not address this, so the full paper needs to make the setup explicit.

This is for people who build or analyze GNNs with positional encodings and care about the gap between theory and the truncated versions everyone actually runs. It is worth sending to referees because the question is practical and the core theoretical distinction looks worth checking, even if the truncation fairness argument needs tightening.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that complete spectral and walk-based positional encodings (PEs) are equivalent in expressive power (between 1-WL and 3-WL), but their truncated versions (first k eigenspaces vs. first k powers of A) are not; specifically, truncated spectral PEs drop to at most 1-WL power. It further distinguishes even closely related spectral families such as k-harmonic distances and shows empirically that mixing truncated PEs outperforms any single family on real-world datasets.

Significance. If the distinctions survive scrutiny, the work supplies the first systematic comparison of truncation effects on PE expressivity and supplies a concrete reason why practitioners should not treat all truncated PEs as interchangeable. The experimental finding that mixtures help is practically useful; the theoretical separation between families under truncation is a clear advance over the existing complete-PE equivalence results.

major comments (3)

[§3] §3 (theoretical analysis of truncated spectral PEs): the reduction to ≤1-WL relies on the claim that the k-truncated Laplacian eigenvectors supply no more distinguishing power than degree sequences. The argument appears to treat the truncation parameter k as directly comparable across families, yet the skeptic correctly notes that k eigenvectors (each a global vector) and k walk powers (local polynomials) do not necessarily deliver equivalent information budgets to a downstream message-passing GNN; a counter-example or explicit embedding-dimension normalization is needed to make the expressivity ordering load-bearing.
[§4] §4 (k-harmonic distances): the separation between truncated Laplacian eigenspaces and k-harmonic distances is presented as evidence that even closely related spectral families diverge under truncation. The proof sketch should explicitly state the GNN architecture and the precise WL test used after the PE is concatenated; without that, it is unclear whether the observed difference is intrinsic or an artifact of how the PE vectors are fed into the same update function.
[Experiments] Experimental section (real-world datasets): the claim that 'a mix of truncated PEs is preferable' is supported only by aggregate accuracy numbers. The paper should report per-dataset ablation tables that isolate the contribution of each family at the same k and the same total embedding dimension; otherwise the mixture benefit could be explained by increased feature dimensionality rather than complementary expressivity.

minor comments (2)

[§2] Notation: the manuscript alternates between 'truncated to k' and 'first k' without a single definition; a short paragraph in §2 clarifying the exact matrix or vector that is retained for each family would remove ambiguity.
[Figures] Figure captions: several plots compare different PE families but do not state the fixed k or the GNN depth used; adding these parameters to every caption would improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript to strengthen the theoretical and experimental sections.

read point-by-point responses

Referee: [§3] §3 (theoretical analysis of truncated spectral PEs): the reduction to ≤1-WL relies on the claim that the k-truncated Laplacian eigenvectors supply no more distinguishing power than degree sequences. The argument appears to treat the truncation parameter k as directly comparable across families, yet the skeptic correctly notes that k eigenvectors (each a global vector) and k walk powers (local polynomials) do not necessarily deliver equivalent information budgets to a downstream message-passing GNN; a counter-example or explicit embedding-dimension normalization is needed to make the expressivity ordering load-bearing.

Authors: We agree that direct comparison of truncation parameter k requires normalization. In the revision we will fix the total PE embedding dimension to the same value d across all families and add a concrete counter-example showing that, even at equal dimension, truncated spectral PEs fail to distinguish certain non-isomorphic graphs that truncated walk-based PEs can distinguish. This will make the ≤1-WL claim rigorous. revision: yes
Referee: [§4] §4 (k-harmonic distances): the separation between truncated Laplacian eigenspaces and k-harmonic distances is presented as evidence that even closely related spectral families diverge under truncation. The proof sketch should explicitly state the GNN architecture and the precise WL test used after the PE is concatenated; without that, it is unclear whether the observed difference is intrinsic or an artifact of how the PE vectors are fed into the same update function.

Authors: We will revise the proof sketch in §4 to explicitly state the GNN architecture (standard MPNN with sum aggregation and MLP updates) and the WL test (1-WL) used after PE concatenation. This will confirm the separation is intrinsic to the truncated families. revision: yes
Referee: [Experiments] Experimental section (real-world datasets): the claim that 'a mix of truncated PEs is preferable' is supported only by aggregate accuracy numbers. The paper should report per-dataset ablation tables that isolate the contribution of each family at the same k and the same total embedding dimension; otherwise the mixture benefit could be explained by increased feature dimensionality rather than complementary expressivity.

Authors: We agree that aggregate numbers are insufficient. The revision will add per-dataset ablation tables comparing each family and their mixtures at identical k and identical total embedding dimension, isolating the contribution of complementary expressivity. revision: yes

Circularity Check

0 steps flagged

No circularity; theoretical distinctions derived independently

full rationale

The paper's core claims rest on new theoretical analysis of truncated positional encodings under the WL hierarchy. The abstract and provided context show no self-definitional reductions, no fitted parameters renamed as predictions, and no load-bearing self-citations that collapse the argument to prior unverified work by the same authors. The distinction between spectral and walk-based truncations, and the corollary that truncated spectral PEs fall to 1-WL, are presented as fresh results rather than rearrangements of inputs. The uniform-k assumption is an explicit modeling choice whose consequences are analyzed, not smuggled in via definition or citation chain. This is a standard case of a self-contained theoretical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work relies on standard background from spectral graph theory and the Weisfeiler-Lehman hierarchy but introduces no new free parameters, invented entities, or ad-hoc axioms visible in the summary.

axioms (1)

standard math Standard definitions and properties of the Weisfeiler-Lehman hierarchy and spectral graph theory hold.
The claims about expressive power are framed in terms of the 1-WL test and spectral encodings, which presuppose these background results.

pith-pipeline@v0.9.1-grok · 5775 in / 1293 out tokens · 30406 ms · 2026-06-27T07:07:13.160080+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages

[1]

Black, M., Lin, L., Wong, W.-K., and Nayyeri, A

URL https://openreview.net/forum? id=i80OPhOCVH2. Black, M., Lin, L., Wong, W.-K., and Nayyeri, A. Bi- harmonic distance of graphs and its higher-order vari- ants: Theoretical properties with applications to cen- trality and clustering. InForty-first International Con- ference on Machine Learning, 2024a. URL https: //openreview.net/forum?id=3pxMIjB9QK. Bl...

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Jambulapati, A

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Johnson and Joram Lindenstrauss

URL https://api.semanticscholar. org/CorpusID:17184843. Johnson, W. and Lindenstrauss, J. Extensions of lipschitz maps into a hilbert space.Contemporary Mathematics, 26:189–206, 01 1984. doi: 10.1090/conm/026/737400. Klein, D. J. and Randi´c, M. Resistance distance.Journal of Mathematical Chemistry, 12(1):81–95, Dec 1993. ISSN 1572-8897. doi: 10.1007/BF01...

work page doi:10.1090/conm/026/737400 1984
[4]

Lim, D., Robinson, J

SIAM, 2018. Lim, D., Robinson, J. D., Zhao, L., Smidt, T., Sra, S., Maron, H., and Jegelka, S. Sign and basis invariant networks for spectral graph representation learning. InThe Eleventh International Conference on Learning Representations,

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Ma, L., Lin, C., Lim, D., Romero-Soriano, A., Dokania, P

URL https://openreview.net/forum? id=Q-UHqMorzil. Ma, L., Lin, C., Lim, D., Romero-Soriano, A., Dokania, P. K., Coates, M., Torr, P., and Lim, S.-N. Graph in- ductive biases in transformers without message passing. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.),Proceedings of the 40th International Conference on...

work page doi:10.1016/0166-218x(84)90113-6 2023
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Rampasek, L., Galkin, M., Dwivedi, V

URL https://openreview.net/forum? id=S1ldO2EFPr. Rampasek, L., Galkin, M., Dwivedi, V . P., Luu, A. T., Wolf, G., and Beaini, D. Recipe for a general, pow- erful, scalable graph transformer. In Oh, A. H., Agarwal, A., Belgrave, D., and Cho, K. (eds.),Advances in Neural Information Processing Systems, 2022. URL https: //openreview.net/forum?id=lMMaNf6oxKM....

Pith/arXiv arXiv 2022
[7]

Zhou, J., Zhou, C., Wang, X., Li, P., and Zhang, M

URL https://openreview.net/forum? id=kmugaw9Kfq. Zhou, J., Zhou, C., Wang, X., Li, P., and Zhang, M. Towards stable, globally expressive graph representations with laplacian eigenvectors.arXiv preprint arXiv:2410.09737, 2024. Zopf, M. 1-wl expressiveness is (almost) all you need, 2022. URLhttps://arxiv.org/abs/2202.10156. 11 Understanding Truncated Positi...

Pith/arXiv arXiv 2024
[8]

We construct a pair of squared Euclidean distances matrices and prove they are indistinguishable by the WL test
[9]

We prove that these distance matrices are the effective resistance matrices of a pair of weighted graphs
[10]

We are able to prove Item 1 analytically

We prove that these two graphs are indistinguishable by the WL test on their weighted adjacency matrices. We are able to prove Item 1 analytically. For Items 2 and 3, we use symbolic software to perform a computer-aided proof. These steps involving taking a matrix pseudoinverse, which can be hard to reason about mathematically but can by checked with a si...

2017
[11]

and would prove that resistance-WL is strong than shortest-path-distance-WL. Conjecture B.2.There exist a pair of unweighted graphs G and H such that G and H are indistinguishable by the resistance WL test, but they are distinguishable by the shortest-path-distance WL test. C. Truncated Walk Encodings Lemma C.1((Lehman et al., 2017, Corollary 10.3.3)).Let...

2017
[12]

Nodes that are distancer < kfrom a leaf of a tree The k-hop neighborhoods of these nodes are the node connected to a path of lengthr(the path connecting the node to the leaf) and a path of lengthk
[13]

Nodes that are distancer < kfrom the root The k-hop neighborhoods of these nodes are the node connected to a path of lengthkand a path of lengthrconnected to two paths of lengthk−r
[14]

Nodes that are distance> kfrom both a leaf and the root The k-hop neighborhood of these nodes are the node con- nected to two paths of lengthk. As k <⌊ n 8 ⌋, there are no nodes of that are both distance < k to a leaf and a root, as the distance between any leaf and a root in either tree is n 4 For any 0< r < k , in both graphs, there are three nodes of d...

arXiv 2011

[1] [1]

Black, M., Lin, L., Wong, W.-K., and Nayyeri, A

URL https://openreview.net/forum? id=i80OPhOCVH2. Black, M., Lin, L., Wong, W.-K., and Nayyeri, A. Bi- harmonic distance of graphs and its higher-order vari- ants: Theoretical properties with applications to cen- trality and clustering. InForty-first International Con- ference on Machine Learning, 2024a. URL https: //openreview.net/forum?id=3pxMIjB9QK. Bl...

work page doi:10.1007/bf01270385 2020

[2] [2]

Jambulapati, A

URL https://openreview.net/forum? id=xAqcJ9XoTf. Jambulapati, A. and Sidford, A. Ultrasparse ultrasparsifiers and faster laplacian system solvers.ACM Transactions on Algorithms, 21(3):1–49, 2025. Jameson, G. J. O. Counting zeros of generalised poly- nomials: Descartes’ rule of signs and laguerre’s ex- tensions.The Mathematical Gazette, 90:223 – 234,

2025

[3] [3]

Johnson and Joram Lindenstrauss

URL https://api.semanticscholar. org/CorpusID:17184843. Johnson, W. and Lindenstrauss, J. Extensions of lipschitz maps into a hilbert space.Contemporary Mathematics, 26:189–206, 01 1984. doi: 10.1090/conm/026/737400. Klein, D. J. and Randi´c, M. Resistance distance.Journal of Mathematical Chemistry, 12(1):81–95, Dec 1993. ISSN 1572-8897. doi: 10.1007/BF01...

work page doi:10.1090/conm/026/737400 1984

[4] [4]

Lim, D., Robinson, J

SIAM, 2018. Lim, D., Robinson, J. D., Zhao, L., Smidt, T., Sra, S., Maron, H., and Jegelka, S. Sign and basis invariant networks for spectral graph representation learning. InThe Eleventh International Conference on Learning Representations,

2018

[5] [5]

Ma, L., Lin, C., Lim, D., Romero-Soriano, A., Dokania, P

URL https://openreview.net/forum? id=Q-UHqMorzil. Ma, L., Lin, C., Lim, D., Romero-Soriano, A., Dokania, P. K., Coates, M., Torr, P., and Lim, S.-N. Graph in- ductive biases in transformers without message passing. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.),Proceedings of the 40th International Conference on...

work page doi:10.1016/0166-218x(84)90113-6 2023

[6] [6]

Rampasek, L., Galkin, M., Dwivedi, V

URL https://openreview.net/forum? id=S1ldO2EFPr. Rampasek, L., Galkin, M., Dwivedi, V . P., Luu, A. T., Wolf, G., and Beaini, D. Recipe for a general, pow- erful, scalable graph transformer. In Oh, A. H., Agarwal, A., Belgrave, D., and Cho, K. (eds.),Advances in Neural Information Processing Systems, 2022. URL https: //openreview.net/forum?id=lMMaNf6oxKM....

Pith/arXiv arXiv 2022

[7] [7]

Zhou, J., Zhou, C., Wang, X., Li, P., and Zhang, M

URL https://openreview.net/forum? id=kmugaw9Kfq. Zhou, J., Zhou, C., Wang, X., Li, P., and Zhang, M. Towards stable, globally expressive graph representations with laplacian eigenvectors.arXiv preprint arXiv:2410.09737, 2024. Zopf, M. 1-wl expressiveness is (almost) all you need, 2022. URLhttps://arxiv.org/abs/2202.10156. 11 Understanding Truncated Positi...

Pith/arXiv arXiv 2024

[8] [8]

We construct a pair of squared Euclidean distances matrices and prove they are indistinguishable by the WL test

[9] [9]

We prove that these distance matrices are the effective resistance matrices of a pair of weighted graphs

[10] [10]

We are able to prove Item 1 analytically

We prove that these two graphs are indistinguishable by the WL test on their weighted adjacency matrices. We are able to prove Item 1 analytically. For Items 2 and 3, we use symbolic software to perform a computer-aided proof. These steps involving taking a matrix pseudoinverse, which can be hard to reason about mathematically but can by checked with a si...

2017

[11] [11]

and would prove that resistance-WL is strong than shortest-path-distance-WL. Conjecture B.2.There exist a pair of unweighted graphs G and H such that G and H are indistinguishable by the resistance WL test, but they are distinguishable by the shortest-path-distance WL test. C. Truncated Walk Encodings Lemma C.1((Lehman et al., 2017, Corollary 10.3.3)).Let...

2017

[12] [12]

Nodes that are distancer < kfrom a leaf of a tree The k-hop neighborhoods of these nodes are the node connected to a path of lengthr(the path connecting the node to the leaf) and a path of lengthk

[13] [13]

Nodes that are distancer < kfrom the root The k-hop neighborhoods of these nodes are the node connected to a path of lengthkand a path of lengthrconnected to two paths of lengthk−r

[14] [14]

Nodes that are distance> kfrom both a leaf and the root The k-hop neighborhood of these nodes are the node con- nected to two paths of lengthk. As k <⌊ n 8 ⌋, there are no nodes of that are both distance < k to a leaf and a root, as the distance between any leaf and a root in either tree is n 4 For any 0< r < k , in both graphs, there are three nodes of d...

arXiv 2011