arxiv: 2604.15069 · v1 · submitted 2026-04-16 · 💻 cs.LG

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Beyond the Laplacian: Doubly Stochastic Matrices for Graph Neural Networks

Zhaobo Hu , Vincent Gauthier , Mehdi Naima

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Pith reviewed 2026-05-10 11:52 UTC · model grok-4.3

classification 💻 cs.LG

keywords graph neural networksdoubly stochastic matricesover-smoothingDirichlet energymodified LaplacianNeumann seriesresidual compensation

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The pith

Substituting the Laplacian with a doubly stochastic matrix from its modified inverse lets GNNs encode multi-hop proximity while bounding Dirichlet energy decay.

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

The paper proposes to replace the Laplacian matrix conventionally used for message passing in graph neural networks with a doubly stochastic matrix derived from the inverse of a modified Laplacian. This matrix is designed to encode continuous multi-hop proximity relations and to enforce strict local centrality. Because exact inversion is expensive, the authors approximate it with a truncated Neumann series and compensate for lost probability mass to keep the matrix row-stochastic. Theoretical analysis shows the approach bounds Dirichlet energy decay across layers, thereby reducing over-smoothing, while experiments confirm efficiency and accuracy on homophilic graph tasks.

Core claim

Substituting the traditional Laplacian with a Doubly Stochastic graph Matrix (DSM), derived from the inverse of the modified Laplacian, naturally encodes continuous multi-hop proximity and strict local centrality. To overcome the intractable O(n^3) complexity of exact matrix inversion, a truncated Neumann series scalably approximates the DSM for DsmNet, while DsmNet-compensate adds a Residual Mass Compensation mechanism that re-injects truncated tail mass into self-loops to restore row-stochasticity. The decoupled architectures operate in O(K|E|) time, mitigate over-smoothing by bounding Dirichlet energy decay, and receive robust empirical validation on homophilic benchmarks, while also su

What carries the argument

The Doubly Stochastic Matrix (DSM) obtained from the inverse of the modified Laplacian and approximated by a truncated Neumann series with residual mass compensation.

Load-bearing premise

The approach rests on the premise that the inverse of the modified Laplacian encodes multi-hop proximity and local centrality in a form that the Neumann truncation and residual compensation can preserve without major distortion.

What would settle it

If Dirichlet energy in the DSM-based GNN decays at the same rate as in a standard Laplacian GNN when both are run for many layers on the same homophilic dataset, the over-smoothing mitigation would be falsified.

Figures

Figures reproduced from arXiv: 2604.15069 by Mehdi Naima, Vincent Gauthier, Zhaobo Hu.

**Figure 2.** Figure 2: t-SNE visualization of the learned node representations on the homophilic Cora dataset. The latent spaces [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Empirical verification of the Neumann series approximation. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Node centrality rank preservation analysis using Kendall’s Tau ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Distance decay of the exact doubly stochastic graph matrix [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Dirichlet energy decay trajectories across four distinct graph topologies (BA, ER, WS, and SBM, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

read the original abstract

Graph Neural Networks (GNNs) conventionally rely on standard Laplacian or adjacency matrices for structural message passing. In this work, we substitute the traditional Laplacian with a Doubly Stochastic graph Matrix (DSM), derived from the inverse of the modified Laplacian, to naturally encode continuous multi-hop proximity and strict local centrality. To overcome the intractable $O(n^3)$ complexity of exact matrix inversion, we first utilize a truncated Neumann series to scalably approximate the DSM, which serves as the foundation for our proposed DsmNet. Furthermore, because algebraic truncation inherently causes probability mass leakage, we introduce DsmNet-compensate. This variant features a mathematically rigorous Residual Mass Compensation mechanism that analytically re-injects the truncated tail mass into self-loops, strictly restoring row-stochasticity and structural dominance. Extensive theoretical and empirical analyses demonstrate that our decoupled architectures operate efficiently in $O(K|E|)$ time and effectively mitigate over-smoothing by bounding Dirichlet energy decay, providing robust empirical validation on homophilic benchmarks. Finally, we establish the theoretical boundaries of the DSM on heterophilic topologies and demonstrate its versatility as a continuous structural encoding for Graph Transformers.

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

The paper swaps the Laplacian for a doubly stochastic matrix in GNNs using Neumann truncation plus diagonal mass compensation, but the abstract gives no proof that the approximation preserves the claimed encodings or bounds.

read the letter

The main takeaway is a concrete substitution: build a doubly stochastic matrix from the inverse of a modified Laplacian, approximate it with a truncated Neumann series for O(K|E|) cost, then analytically push the leaked mass back onto self-loops to restore row-stochasticity. They position this as DsmNet and DsmNet-compensate, with claims that it encodes multi-hop proximity better, bounds Dirichlet energy decay to limit over-smoothing, and has theoretical limits on heterophilic graphs. The compensation step is presented as rigorous and structure-preserving. That combination of DSM construction, truncation, and re-injection is the actual novelty here; I have not seen this exact recipe before. The efficiency argument and the motivation around continuous structural encodings are laid out clearly. The soft spot is exactly the one in the stress-test note. The abstract never shows the algebra confirming that diagonal-only compensation keeps column-stochasticity, effective-resistance distances, or the original centrality scores intact. If those properties shift, the Dirichlet bound and the heterophily claims lose their foundation. No equations, no proof sketches, and no error analysis appear in the provided text, so the theoretical sections cannot be checked yet. Experiments are mentioned only for homophilic benchmarks with no numbers or ablation details. This is for people who work on propagation matrices and over-smoothing fixes. A reader already thinking about alternatives to the normalized Laplacian could extract a usable idea, but the paper is not yet at the point where I would cite it. It deserves a serious referee because the core substitution is fresh and the runtime claim is testable; the authors should be asked to supply the missing derivations and broader experiments.

Referee Report

2 major / 2 minor

Summary. The paper proposes replacing conventional Laplacian or adjacency matrices in GNNs with a Doubly Stochastic graph Matrix (DSM) obtained from the inverse of a modified Laplacian. This DSM is claimed to encode continuous multi-hop proximity and strict local centrality. Scalability is achieved via a truncated Neumann series approximation yielding O(K|E|) complexity for the proposed DsmNet; a Residual Mass Compensation step analytically re-injects truncated tail mass into self-loops to restore row-stochasticity, producing DsmNet-compensate. The authors assert that the resulting architectures bound Dirichlet energy decay to mitigate over-smoothing, supply theoretical limits on heterophilic graphs, and serve as a continuous structural encoding for Graph Transformers, with supporting empirical results on homophilic benchmarks.

Significance. If the DSM construction and its Neumann approximation plus compensation provably preserve the claimed multi-hop proximity, local centrality, and effective-resistance properties while delivering the stated Dirichlet-energy bound, the approach would supply a new, theoretically grounded structural operator for message passing that directly targets over-smoothing. The O(K|E|) complexity and extension to Graph Transformers would be practically useful strengths.

major comments (2)

[Abstract, §3] Abstract and §3 (DSM derivation and compensation): the claim that the Residual Mass Compensation 'strictly restoring row-stochasticity and structural dominance' preserves the DSM's multi-hop proximity encodings and local centrality without distortion is load-bearing for both the Dirichlet-energy bound and the heterophily analysis. Adding mass exclusively to the diagonal restores row sums but does not automatically guarantee column-stochasticity or invariance of effective-resistance distances; a concrete proof or counter-example showing that centrality scores and the energy bound remain intact after truncation is required.
[§4] §4 (theoretical analysis): the asserted bounds on Dirichlet energy decay and the 'theoretical boundaries of the DSM on heterophilic topologies' are stated without displayed equations or proof sketches in the provided text. Because these bounds are used to justify over-smoothing mitigation and heterophily claims, the derivation steps (including how the modified Laplacian inverse yields the stated properties) must be supplied explicitly.

minor comments (2)

[§2] Notation for the modified Laplacian and the exact definition of the DSM (e.g., whether it is L^{-1} normalized or directly inverted) should be introduced with an equation in §2 or §3 for clarity.
[§5] The empirical section would benefit from reporting variance across multiple random seeds and a direct comparison against recent heterophily-specific baselines (e.g., H2GCN, GPR-GNN) rather than only homophilic benchmarks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects of the theoretical justification for the DSM construction and its approximations. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the proofs and derivations.

read point-by-point responses

Referee: [Abstract, §3] Abstract and §3 (DSM derivation and compensation): the claim that the Residual Mass Compensation 'strictly restoring row-stochasticity and structural dominance' preserves the DSM's multi-hop proximity encodings and local centrality without distortion is load-bearing for both the Dirichlet-energy bound and the heterophily analysis. Adding mass exclusively to the diagonal restores row sums but does not automatically guarantee column-stochasticity or invariance of effective-resistance distances; a concrete proof or counter-example showing that centrality scores and the energy bound remain intact after truncation is required.

Authors: We agree that explicit verification of property preservation after compensation is necessary for the load-bearing claims. The exact DSM is the inverse of a symmetric modified Laplacian and is therefore doubly stochastic. The truncated Neumann series preserves symmetry when applied to a symmetric base matrix. The residual mass (the per-row deficit after truncation) is added only to the diagonal to enforce exact row-stochasticity. We will insert a new theorem in §3 proving that the compensated matrix perturbs the original effective-resistance distances by at most O(1/K) and that eigenvector centrality scores remain continuous under this diagonal perturbation, thereby preserving the multi-hop proximity encoding up to a controllable error. The Dirichlet-energy bound will be restated with an additive term that vanishes as K increases. Column-stochasticity holds exactly for the untruncated DSM and approximately for the compensated version, with the deviation bounded by the tail of the Neumann series. These additions will be included in the revision. revision: yes
Referee: [§4] §4 (theoretical analysis): the asserted bounds on Dirichlet energy decay and the 'theoretical boundaries of the DSM on heterophilic topologies' are stated without displayed equations or proof sketches in the provided text. Because these bounds are used to justify over-smoothing mitigation and heterophily claims, the derivation steps (including how the modified Laplacian inverse yields the stated properties) must be supplied explicitly.

Authors: We acknowledge that the derivation steps in §4 were presented too concisely. The Dirichlet-energy bound follows from showing that the DSM operator is a contraction on the subspace orthogonal to the all-ones vector, with contraction rate given by 1 minus the second-smallest eigenvalue of the modified Laplacian; the energy therefore decays geometrically. For heterophilic graphs we bound the label-propagation distance by relating the continuous proximity values in the DSM to the graph's Cheeger constant. We will expand §4 with the full sequence of lemmas, the explicit energy-decay inequality, and the heterophily boundary derivation, including all intermediate equations. These clarifications will be added without changing the stated claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from matrix definitions and standard approximations

full rationale

The paper defines the DSM explicitly as the inverse of a modified Laplacian, introduces the truncated Neumann series as a standard scalable approximation technique, and describes the residual mass compensation as an analytical re-injection into self-loops that restores row-stochasticity by direct construction. None of these steps reduce a claimed prediction, uniqueness result, or theoretical bound to a fitted parameter, self-citation chain, or input by definition. The Dirichlet energy bound for over-smoothing mitigation follows from the DSM properties rather than being presupposed. No load-bearing step matches any of the enumerated circularity patterns, consistent with the default expectation that most papers are non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Review limited to abstract; key elements are the derivation of DSM and the validity of the series approximation plus compensation. No fitted parameters are mentioned.

axioms (2)

standard math The Neumann series provides a valid scalable approximation to the inverse of the modified Laplacian.
Invoked to achieve O(K|E|) complexity instead of O(n^3) inversion.
domain assumption Re-injecting truncated tail mass into self-loops restores row-stochasticity without distorting the intended multi-hop proximity encoding.
Central to the DsmNet-compensate variant described in the abstract.

invented entities (1)

Doubly Stochastic graph Matrix (DSM) no independent evidence
purpose: To encode continuous multi-hop proximity and strict local centrality as a replacement for the Laplacian in message passing.
Derived from modified Laplacian inverse; presented as the core structural encoding.

pith-pipeline@v0.9.0 · 5498 in / 1452 out tokens · 38495 ms · 2026-05-10T11:52:10.967822+00:00 · methodology

discussion (0)

Reference graph

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