Demystifying MPNNs: Message Passing as Merely Efficient Matrix Multiplication

Chengjia Wang; Michael Lones; Qin Jiang; Wei Pang

REVIEW 4 major objections 6 minor 1 cited by

Demystifying MPNNs: Message Passing as Merely Efficient Matrix Multiplication

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Message passing is just efficient matrix multiplication

desk verdict The central A^k equivalence is unsupported (invalid UAT step, concrete counterexample), but the empirical depth-vs-neighborhood decomposition and loop analysis deserve referee time. read the letter →

arxiv 2502.00140 v1 pith:J6JZGOYB submitted 2025-01-31 cs.LG cs.AIcs.NEcs.SI

Qin Jiang , Chengjia Wang , Michael Lones , Wei Pang This is my paper · ORCID

classification cs.LGcs.AIcs.NEcs.SI

keywords graphneuralnetworksmessagepassingadjacencymatrixpowersk-hopneighborhoodsover-smoothingnodedegreeuniversalapproximationclassification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that message passing in graph neural networks is fundamentally repeated matrix multiplication by the adjacency matrix. The central claim is that a $k$-layer GCN without self-loops aggregates only the $k$-hop neighborhood, not a cumulative 1-through-$k$ neighborhood, and behaves approximately like a single-layer GCN on the matrix power $A^k$. This reframing makes message passing a memory-efficient way to compute $A^k$ times the feature matrix, and it has direct consequences: adding self-loops or symmetrizing graphs mixes hops and drives over-smoothing, while in sparse directed graphs deep-network failures can come from gradients instead. The paper also shows that with uniform node features, degree becomes the effective feature, making structure-only tasks a special case of hybrid tasks.

What carries the argument

The load-bearing object is the $k$-th power of the adjacency matrix, $A^k$, whose nonzero entries count directed paths of length exactly $k$. The identity $H^{(k)} = \sigma((W \odot A)^k X W^{(k)})$ carries the argument: it says depth in an MPNN computes powers of the normalized adjacency matrix. The proof's enabling step is the binomial expansion $(A+I)^k = \sum_{i=0}^k \binom{k}{i} A^i$, which converts the self-loop formula into a sum of $k$-hop features. The path-counting interpretation of matrix powers, proved by induction in Lemma 2.5, connects algebraic powers to graph-theoretic neighborhoods.

What would settle it

Train a $k$-layer GCN without self-loops and, in parallel, a single-layer GCN whose adjacency matrix is $A^k$, using the same weight initialization and training budget on a small graph; if the two models' outputs or test accuracies differ substantially (beyond what function-approximation error would allow), the claimed equivalence is falsified, because the closed form predicts they should be interchangeable.

Watch

Extended reading notes

Core claim

The paper's central discovery is the closed-form equivalence expressed in Lemma 2.7: for a $k$-layer GCN without self-loops, $H^{(k)} = \sigma((W \odot A)^k X W^{(k)})$, where $W$ is the normalization weight matrix. The proof proceeds by induction and uses the Universal Approximation Theorem to justify dropping the activation inside each layer. For a GCN with self-loops, the analogous formula gives $\sigma((W \odot (A+I))^k X W^{(k)})$, which by the binomial theorem is a linear combination of all powers $A^0$ through $A^k$; GraphSAGE produces a similar combination through its separate self-feature path. The paper also proves loop-driven inheritance rules (self-loops add one hop, undirected edges add two, $m$-cycles add $m$) and uses them to explain over-smoothing and to identify gradient-related degradation in sparse graphs. For uniform features, it shows a $(k+1)$-layer network equals a $k$-layer network using node degrees as features.

Load-bearing premise

The proof that a $k$-layer network equals a single layer on $A^k$ assumes that removing the nonlinear activation inside each layer, justified by the Universal Approximation Theorem, does not change the network's behavior; the theorem only guarantees that some network can approximate a function, not that deleting this specific activation is harmless.

Editorial extensions

If this is right

A $k$-layer GCN without self-loops uses exactly the $k$-hop neighbors of each node, not a cumulative aggregation of all shorter hops.
Because $(A+I)^k$ expands into all powers of $A$, adding self-loops (or working on undirected graphs) makes $k$-hop neighborhoods contain shorter-hop neighbors, which the paper identifies as a mechanism for over-smoothing.
Deep GCNs can maintain accuracy up to 50 layers on sparse directed graphs when self-loops and undirected symmetrization are removed, indicating gradient problems rather than over-smoothing are the main failure mode there.
With uniform node features, node degree becomes the effective feature: a $(k+1)$-layer MPNN with constant features is equivalent to a $k$-layer MPNN using degree as the only feature.
Message passing is a memory-efficient alternative to explicitly forming $A^k$ and multiplying it with $X$, since iterative multiplication avoids storing the dense power matrix.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central equivalence is accepted, then tuning depth is effectively tuning the power of the adjacency matrix; this suggests that layer count and neighborhood radius are interchangeable up to memory constraints, and that residual or normalization tricks that preserve the linear form may be more effective than architectural modifications aimed at over-smoothing.
The paper's distinction between density-driven over-smoothing (loops, undirected edges) and gradient-driven degradation (sparse directed graphs) could be turned into a diagnostic: compare the $k$-layer model's accuracy with the single-layer $A^k$ model's accuracy; a large gap with comparable neighborhoods points to gradient or optimization issues rather than over-smoothing.
A testable extension: on graphs where adding self-loops hurts, the theory predicts the harm scales with the density of $(A+I)^k$ versus $A^k$; this can be checked by ablating self-loops on a range of homophily and heterophily datasets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper's central theoretical result is not proven. The step in Appendix A.2 (Eq. 5 to Eq. 6) where they drop the inner nonlinearity 'thanks to UAT' is invalid, and a simple two-layer ReLU GCN does not match the claimed A^k representation. So the 'message passing is merely efficient matrix multiplication' framing, at least as an equivalence, does not hold for nonlinear MPNNs.

That said, the paper does some things well. The experimental design in Figure 4 (growing layers vs growing k vs k-hop plus linear layers) is a genuinely useful way to separate gradient degradation from oversmoothing, and the result that a directed GCN without self-loops stays stable up to 50 layers on Chameleon/Squirrel is a real observation worth explaining. The loop-coexistence lemmas (self-loops extend k-hop to k+1, undirected edges extend to k+2) are elementary but correct, and they do clarify when repeated powers of A create dense connectivity. The structure-only analysis—uniform features reduce to degree counting, row normalization destroys degree information—is correct and cleanly explained.

The soft spots are serious but located. The proof of Lemma 2.7/2.8 is the load-bearing wall. UAT does not license removing a fixed nonlinearity from a fixed compositional network; it says some network of sufficient width can approximate a target function. The stress-test counterexample (a specific 6-node graph where 2-layer GCN output at node 6 is 2 but the A^2 single-layer is 1) is a clean demonstration. Also, the notation (W ⊙ A)^k requires either weight sharing across layers or an unjustified absorption of per-layer weights; as written it is not the adjacency matrix power. The paper also occasionally conflates 'approximation' with 'exact equivalence.' The experiments are interesting but they are used to support the flawed theory, so the interpretation of Figures 4-6 needs to be re-framed once the theory is corrected.

Minor: Appendix A.4's conclusion line for GraphSAGE accidentally writes the GCN-with-self-loop formula, which is a typo but symptomatic of haste.

Bottom line: this paper is a mix of a failed central proof and some genuinely useful empirical/observational work. Researchers working on GNN depth and oversmoothing will find the experiments thought-provoking, but they should not rely on the theoretical equivalence. A serious reviewer could help the authors fix or drastically weaken the theoretical claims and keep the experiments. I'd send it to review, not desk-reject, because the experimental contribution and the loop analysis are worth engaging with. I wouldn't cite the equivalence, but I might cite the Figure 4 methodology or the directed no-self-loop stability finding if the authors re-scope the paper.

Referee Report

4 major / 6 minor

Summary. The paper aims to provide a theoretical account of message-passing neural networks (MPNNs), claiming three main results: (1) a k-layer MPNN operating on adjacency matrix A is approximately equivalent to a single-layer MPNN operating on A^k, so that message passing is essentially a memory-efficient way to compute powers of the adjacency matrix; (2) different loop structures (self-loops, two-node loops, longer cycles) cause hop coexistence and thereby contribute to over-smoothing; and (3) node classification tasks can be split into structure-feature hybrid and structure-only types, with the latter reducible to degree-based feature prediction. The theoretical core is Lemma 2.7 (and Lemmas 2.8, 2.9), whose proofs in Appendix A rely on removing the inner nonlinearity via the Universal Approximation Theorem. The empirical sections compare deep GCNs, single-layer GCNs on k-hop neighborhoods, and hybrid architectures on citation and web graphs, and they study normalization schemes with uniform features.

Significance. If the central equivalence were valid, the paper would provide a clean and useful interpretation of deep GNNs as iterative matrix-power computation, and it would connect loop structure to over-smoothing through hop coexistence. The empirical comparison of depth growth versus k-hop neighborhood growth on directed sparse graphs is a useful experiment, and the observation that row normalization with uniform features destroys degree information is sound. The paper also provides a code link and reports standard deviations over multiple splits. However, the main theoretical claim is not established: the proof of the key lemmas uses an invalid application of the Universal Approximation Theorem, and a concrete ReLU counterexample shows that the claimed A^k equivalence is false as stated. The paper's central theoretical contribution is therefore unsupported, even though some of its empirical observations may be of independent interest.

major comments (4)

[Appendix A.2, Eqs. (5)-(6)] The induction step is not a proof. After substituting the induction hypothesis, the expression σ(W⊙(A+I) σ((W⊙(A+I))^k X W^(k)) W^(k+1)) is replaced by σ((W⊙(A+I))^(k+1) X W^(k+1)) with the sole justification 'Thanks to the Universal Approximation Theorem'. UAT asserts that some sufficiently wide network can approximate a continuous function; it does not license deleting a fixed nonlinearity from a fixed compositional expression, nor does it imply that the composition is exactly representable as σ(M X W') for a single matrix M. The same unjustified step is used in Appendix A.3 for Lemma 2.7 and in Appendix A.4, Eqs. (9)-(10), for Lemma 2.9. Since this step is the load-bearing part of the paper's central claim, the proof is invalid.
[Section 2.2, Lemma 2.7] The claimed identity is false for standard ReLU GCNs. Consider the directed graph with edges 1→4, 2→4, 3→5, 4→6, and 5→6, using the paper's convention A_{ij}=1 for an edge i→j, node feature vector X=[1,1,-1,0,0,0]^T, identity weight matrices, and ReLU activation. The 2-layer GCN without self-loops gives output 2 at node 6, while the single-layer GCN on A^2 gives ReLU(1)=1 at node 6. As functions, the two-layer output is ReLU(ReLU(x1+x2)+ReLU(x3)) and the single-layer output is ReLU(x1+x2+x3), which differ on an open set. Thus the A^k equivalence, and with it the 'efficient matrix multiplication' interpretation, is unsupported for nonlinear MPNNs.
[Introduction and Section 2.3] The paper's advertised result is that a k-layer MPNN on adjacency matrix A is approximately equivalent to a single-layer MPNN on A^k, but Lemma 2.7 actually states H^(k)=σ((W⊙A)^k X W^(k)), where W⊙A is the normalization-mask form of the adjacency matrix introduced in Appendix E.1, Eq. (12). For a symmetrically normalized GCN, (D^{-1/2}AD^{-1/2})^k is not A^k, and the distinction affects both the neighborhood interpretation and the experiments in Section 4, which use powers of the original adjacency matrix. The formal statement of the paper's own lemma therefore does not match the headline claim.
[Appendix A.4, Lemma 2.9] The proof of Lemma 2.9 repeats the same invalid UAT step: Eq. (9) contains inner σ terms applied to sums of powers of (W⊙A), and Eq. (10) removes those σ terms with no justification other than the same UAT sentence. In addition, the conclusion line of the proof states H^(n)=σ((W⊙(A+I))^n X W^(n)), which is the GCN-with-self-loop formula rather than the GraphSAGE expression claimed in Lemma 2.9. Since the derivation is central to the paper's treatment of GraphSAGE, this lemma is also unsupported.

minor comments (6)

[Lemma 2.3] The lemma uses 'kth order node feature' but the statement is about p-hop neighbors; the variables p and k should be made consistent.
[Equation (6)] The expression cW^(k+1) contains an unexplained symbol c; this appears to be a typo.
[Notation throughout] The symbol W is overloaded: W⊙A denotes a fixed normalization mask, while W^(l) denotes learnable feature-transformation matrices. These should be distinguished by different symbols, especially in Lemmas 2.7-2.9.
[Section 4.2.1] The statement that row normalization with homogeneous features makes 'all nodes get identical representations' assumes zero biases and identical initial features; the conditions should be stated explicitly.
[Appendix C.2.1] The citation 'Rossi et al. (?)' is incomplete and should be filled in.
[Table 1 and Figures 4-5] The density of the k-hop adjacency matrix is used to support the over-smoothing discussion, but the definition is not precise about whether density counts nonzero entries or path counts, nor about how normalization affects the reported values.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the A^k equivalence is not fit-derived or self-referential; the UAT-based deletion of inner nonlinearities is an unsupported proof step, and the only self-citations are non-load-bearing pointers.

full rationale

The derivation chain contains no step where a claimed prediction is fitted from data or defined in terms of its own target. Lemma 2.7 and 2.8 (Appendix A.2/A.3) are inductive proofs built on the layer update H^(k+1)=σ(W⊙(A+I)H^(k)W^(k)); the closed forms are not definitions of the output, and no fitted constant is used to force them. The one non-algebraic move is Eq. 5 to Eq. 6 in Appendix A.2 (and its parallel in A.3), where UAT is invoked to justify removing the inner σ from the composition. That step is mathematically unsupported—UAT does not license deleting a fixed activation from a fixed layerwise network—and a two-layer ReLU example contradicts the exact A^k equality, so the proof is not reliable as written. Unsupportedness is a correctness flaw, however, not a circular reduction: the conclusion is not an input to the proof. The only self-citations (Jiang et al., 2024) point to hyperparameter settings in the footnote of Section 4.2.1 and to background on higher-order models in E.1/E.2; they do not carry the central A^k argument. The score of 2 reflects that minor non-load-bearing self-citation; no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's theoretical machinery is standard linear algebra plus one non-standard move: using UAT to collapse nonlinearities. No fitted free parameters appear in the theory; the experimental claims rely on dataset-specific hyperparameters. No new entities are introduced.

assumptions (3)

ad hoc to paper Universal Approximation Theorem justifies removing nonlinear activations inside each layer of the unrolled network without changing the function (Eq. 5-6, Appendix A.2).
UAT guarantees existence of some network approximating a target function; it does not imply that a specific layer-wise composition with residual nonlinearities is exactly equivalent to the linearized product. The proof uses it to collapse sigma, which is outside the theorem's scope.
domain assumption Graphs are directed with adjacency entries in non-negative integers; undirected graphs are treated as bidirected directed graphs.
Used throughout Sections 2-3 to define k-hop neighborhoods and matrix powers. The paper treats undirected edges as two directed edges, which is standard but is a modeling choice.
domain assumption The conclusion that gradient issues dominate in sparse graphs generalizes from the tested citation and web datasets.
Section 4.1 draws a general conclusion from CiteSeer, CoraML, and PubMed. The paper itself notes dense networks behave differently (Section 4.1), so the scope is dataset-dependent.

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Cite this review

Pith. "Pith review of Demystifying MPNNs: Message Passing as Merely Efficient Matrix Multiplication." pith.science (2026). https://pith.science/paper/J6JZGOYB

@misc{pith2026250200140,
  author       = {Pith},
  title        = {Pith review of: Demystifying MPNNs: Message Passing as Merely Efficient Matrix Multiplication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6JZGOYB}},
  note         = {Machine review of arXiv:2502.00140}
}

abstract

While Graph Neural Networks (GNNs) have achieved remarkable success, their design largely relies on empirical intuition rather than theoretical understanding. In this paper, we present a comprehensive analysis of GNN behavior through three fundamental aspects: (1) we establish that \textbf{$k$-layer} Message Passing Neural Networks efficiently aggregate \textbf{$k$-hop} neighborhood information through iterative computation, (2) analyze how different loop structures influence neighborhood computation, and (3) examine behavior across structure-feature hybrid and structure-only tasks. For deeper GNNs, we demonstrate that gradient-related issues, rather than just over-smoothing, can significantly impact performance in sparse graphs. We also analyze how different normalization schemes affect model performance and how GNNs make predictions with uniform node features, providing a theoretical framework that bridges the gap between empirical success and theoretical understanding.

Figures

Figures reproduced from arXiv: 2502.00140 by the authors.

**Figure 1.** A k-layer GCN without adding selfloop will only gather information from k-hop neibors. order node feature AkX. Remark 2.4. AkXW is a linear transformation of k-hop neighbor features AkX using weight matrix W. Lemma 2.5. In the k-th power of the adjacency matrix Ak , a non-zero element Ak ij > 0 indicates that there exists at least one directed path of length exactly k from node i to node j. Furthermore, the value of… view at source ↗

**Figure 2.** Types of loops in graphs: (a) self-loop, (b) loop with two nodes connected by an undirected edge, (c) and (d) are examples of n-node loops where n=3 and n=4 respectively. Lemma 2.9. For all natural numbers k, the output of a k-layer GraphSAGE can be expressed as: H(k) = σ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Performance of Unidirectional GCN Without Self-loops on Chameleon and Squirrel Datasets: Model demonstrates stable accuracy up to 50 layers, with deeper architectures constrained by memory limitations. The solid line represents mean accuracy, while the shaded region indicates standard deviation across 10 data splits. 4. Structure-Feature Dichotomy in Node Classification Graph Neural Networks (GNNs) combine node feat… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Comparison of different GCN architectures on three datasets: k-layer GCN (blue), 1-layer GCN with k-hop neighbors (red), and k-hop neighbors with 1-layer GCN and (k-1) linear layers (green). The black line shows the density of k-hop adjacency matrix. menting this find…

**Figure 5.** Figure 5: Comparison of different GCN architectures on CiteSeer dataset under different adjacency matrix formulations. (Top) Using transposed adjacency matrix A T , which propagates information from cited papers to citing papers. (Bottom) Using undirected graph adjacency matrix …

**Figure 6.** Figure 6: On Telegram dataset, layer growth achieves good accuracy while k growth performs poorly with no predictive power. Density shows the percentage of non-zero elements in the equivalent adjacency matrix. Symmetric Normalization Symmetric normalization (Kipf & Welling, 20…

**Figure 7.** Figure 7: Comparison of different GCN architectures on CoraML dataset under different adjacency matrix formulations. (Top) Using transposed adjacency matrix A T , which propagates information from cited papers to citing papers. (Bottom) Using undirected graph adjacency matrix A …

**Figure 8.** Figure 8: Comparison of different GCN architectures on PubMed dataset under different adjacency matrix formulations. (Top) Using transposed adjacency matrix A T , which propagates information from cited papers to citing papers. (Bottom) Using undirected graph adjacency matrix A …

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write newline

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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Graph Neural Networks Use Graphs When They Shouldn 't

Bechler-Speicher, M., Amos, I., Gilad-Bachrach, R., and Globerson, A. Graph Neural Networks Use Graphs When They Shouldn 't. June 2024

work page 2024

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I., Bronstein, M., Webb, S., and Rossi, E

Chamberlain, B., Rowbottom, J., Gorinova, M. I., Bronstein, M., Webb, S., and Rossi, E. GRAND : Graph Neural Diffusion . In Proceedings of the 38th International Conference on Machine Learning , pp.\ 1407--1418. PMLR, July 2021. ISSN: 2640-3498

work page 2021

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Scale-aware Message Passing For Graph Node Classification

Jiang, Q., Wang, C., Lones, M., and Pang, W. Scale Invariance of Graph Neural Networks , December 2024. arXiv:2411.19392 [cs]

work page Pith review arXiv 2024