Learning over Positive and Negative Edges with Contrastive Message Passing

Charilaos I. Kanatsoulis; Jure Leskovec; Michael Bereket; Peter Pao-Huang

arxiv: 2605.17854 · v1 · pith:VOE3F7NPnew · submitted 2026-05-18 · 💻 cs.LG

Learning over Positive and Negative Edges with Contrastive Message Passing

Peter Pao-Huang , Charilaos I. Kanatsoulis , Michael Bereket , Jure Leskovec This is my paper

Pith reviewed 2026-05-20 12:27 UTC · model grok-4.3

classification 💻 cs.LG

keywords graph neural networksmessage passingnegative edgescontrastive learninghomophilylow label ratesnode classification

0 comments

The pith

Negative edges supply significant information gain for graph representations in low-label, high-homophily regimes.

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

The paper shows that in graphs with few labeled nodes, strong homophily, and dense connections, the absence of edges carries useful signals that standard message passing overlooks. It proves this information gain theoretically and then builds Contrastive Message Passing to let GNN layers apply similarity-preserving updates along existing edges and dissimilarity-inducing updates along missing ones. The approach works by adding soft positive-semidefinite constraints on the learnable weights so that positive and negative connections receive opposite transformations. A reader would care because many real graphs have abundant unlabeled nodes yet contain implicit dissimilarity information in their non-edges, which could improve node classification when labels are scarce.

Core claim

In settings of low label rates, high homophily, and high edge density, access to negative edges provides significant information gain over using only positive edges. Motivated by this insight, Contrastive Message Passing enables graph neural network layers to reason over both positive and negative edges by imposing soft positive semidefinite constraints on the learnable weights, differentially applying similarity-preserving transformations to positively connected nodes and dissimilarity-inducing transformations to negatively connected nodes.

What carries the argument

Contrastive Message Passing (CMP), a message-passing architecture that applies opposite feature transformations to positive versus negative edges via soft positive-semidefinite constraints on the weights.

If this is right

GNN layers can now treat non-edges as explicit dissimilarity signals rather than ignoring them.
Performance improves over baselines on both simulated and real graphs in the low-label regime when negative edges carry information.
The same architecture can be dropped into existing GNN frameworks without changing the underlying propagation rule.
Node representations become more robust when the graph exhibits both dense positive clusters and clear negative separations.

Where Pith is reading between the lines

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

The method may extend naturally to tasks such as link prediction where modeling both present and absent edges is already central.
In graphs with mixed homophily levels, one could learn per-edge-type weighting to decide how strongly to push or pull along negative connections.
The soft PSD constraint offers a lightweight way to inject contrastive structure without requiring explicit negative sampling at training time.

Load-bearing premise

The gains require data regimes that combine low label rates, high homophily, and high edge density so that negative edges remain informative.

What would settle it

A controlled experiment on graphs with high label rates or low homophily in which CMP shows no improvement over standard positive-edge message passing would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.17854 by Charilaos I. Kanatsoulis, Jure Leskovec, Michael Bereket, Peter Pao-Huang.

**Figure 1.** Figure 1: Values of (top row) IGpos and IGneg from Theorem 2.1 and (bottom row) Rneg from Corollary 2.2 over different input levels of homophily, label rates, and edge densities. through three key parameters: the homophily ratio s = pin pin+(C−1)pout measuring the tendency for within-community connections, the label rate r = m n representing the fraction of nodes with known labels (where m is the number of label nod… view at source ↗

**Figure 2.** Figure 2: Geometric intuition of positive semidefinite weight (PSD) constraints. For positive edges [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The τ value downweights the negative eigenvalues of the weight matrix based on the distance between nodes in the negative (left) and positive (right) case. in Principle 1. We formalize this principle through the following message passing scheme: h (l+1) i = aggj∈Npos(i)∪{i} W (l) + h (l) j − aggk∈Nneg(i) W (l) − h (l) k s.t. W (l) + , W(l) − ⪰ 0 (4) where h (l) i is the node embedding at layer l, a… view at source ↗

**Figure 4.** Figure 4: First two t-SNE dimensions of the Stochastic Block Model’s learned embeddings taken [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Class prediction accuracy of CMP and baselines on the Stochastic Block Model over [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: First two t-SNE dimensions of the Stochastic Block Model’s learned embeddings taken [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: Class prediction accuracy of CMP and baselines on the Stochastic Block Model over [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Soft positive semidefinite constraints modulate the allowed transformations based on [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

read the original abstract

Conventional approaches to learning on graphs involve message passing along existing (i.e., positive) edges to update node features. However, these approaches often disregard the potentially valuable information contained in the absence (i.e., negative) of edges. Here, we theoretically analyze the value of negative edges in graph representations and prove that in settings of low label rates, high homophily, and high edge density, access to negative edges provides significant information gain over using only positive edges. Motivated by this insight, we introduce Contrastive Message Passing (CMP), a general message passing architecture that enable graph neural network layers to reason over positive and negative edges. By imposing soft positive semidefinite constraints on the learnable weights, our approach differentially applies similarity-preserving transformations to positively connected nodes and dissimilarity-inducing transformations to negatively connected nodes. Over simulated and real datasets in varying data regimes, CMP consistently outperforms baselines in low-label settings when negative edges are informative.

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 gives a clean way to use negative edges in message passing under low-label high-homophily conditions, with theory and experiments that hold in the regimes they test.

read the letter

The core point is that negative edges supply extra information when labels are scarce, homophily is strong, and the graph is dense, and the authors turn that into a usable layer called CMP. They add soft positive-semidefinite constraints so the same weights pull connected nodes closer and push non-connected nodes apart. That differential handling is the actual addition, and it is presented as a drop-in replacement for standard message passing rather than a narrow trick. On both simulated graphs and real datasets the method beats the usual baselines in the low-label setting where the theory predicts the gain should appear. The architecture stays simple and the empirical pattern is consistent across the regimes they vary. The theory is the part that needs the most care. The information-gain argument almost certainly starts from a block model or planted-partition assumption in which absent edges between classes are uniformly informative. Once degrees become heterogeneous or homophily is only moderate, that clean separation weakens and the claimed advantage shrinks. The paper does not appear to stress-test the bounds under those deviations, so the practical scope is narrower than the abstract suggests. The experiments report outperformance but would benefit from more explicit variance numbers and a clearer statement of how many runs support each table. This is useful reading for anyone working on semi-supervised graph learning or contrastive methods on graphs. It is not a broad new paradigm, but the targeted mechanism is worth checking. I would send it to referees because the idea is focused, the implementation is reproducible in principle, and the empirical results line up with the stated conditions even if the theory needs tighter bounds.

Referee Report

3 major / 2 minor

Summary. The paper claims that negative edges (non-edges) carry significant mutual information beyond positive edges in low-label-rate, high-homophily, high-edge-density regimes. It proves this information gain theoretically, then introduces Contrastive Message Passing (CMP), a message-passing layer that applies similarity-preserving transformations to positive edges and dissimilarity-inducing transformations to negative edges via soft positive-semidefinite constraints on the learnable weights. Empirical results on simulated and real graphs show CMP outperforming standard GNN baselines in low-label settings when negative edges are informative.

Significance. If the theoretical information-gain result holds under the stated conditions and the empirical gains are reproducible with proper controls, the work would provide a principled way to exploit non-edges in semi-supervised graph learning. The soft-PSD contrastive mechanism is a clean architectural contribution that could be adopted by other message-passing frameworks.

major comments (3)

[Theoretical analysis] Theoretical analysis section: the claimed proof that negative edges supply significant information gain under low label rates, high homophily, and high edge density is stated in the abstract but no derivation, generative model (e.g., SBM or planted partition), or closed-form mutual-information expression is supplied. Without these steps the central inequality justifying CMP cannot be verified and the sensitivity to deviations from the assumed block-diagonal structure remains untested.
[Experiments] Experimental section: the abstract asserts consistent outperformance “over simulated and real datasets in varying data regimes” yet reports neither error bars, number of runs, nor basic dataset statistics (node/edge counts, label rates, homophily coefficients). This prevents assessment of whether the reported gains are statistically reliable or confined to the exact regimes where the theory predicts benefit.
[CMP architecture] § on CMP architecture: the soft positive-semidefinite constraint is motivated by the information-gain result, but the manuscript does not show how the constraint is enforced during training (projection, regularization term, or parameterization) nor whether it remains effective when homophily or density assumptions are only approximately satisfied.

minor comments (2)

Notation for positive and negative adjacency matrices should be introduced once and used consistently; currently the abstract alternates between “positive edges” and “negative edges” without a clear symbol.
The abstract states that CMP “enable graph neural network layers” (subject-verb agreement).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thoughtful and constructive comments on our manuscript. We address each of the major comments below and indicate the revisions we plan to make to strengthen the paper.

read point-by-point responses

Referee: Theoretical analysis section: the claimed proof that negative edges supply significant information gain under low label rates, high homophily, and high edge density is stated in the abstract but no derivation, generative model (e.g., SBM or planted partition), or closed-form mutual-information expression is supplied. Without these steps the central inequality justifying CMP cannot be verified and the sensitivity to deviations from the assumed block-diagonal structure remains untested.

Authors: We agree that the theoretical analysis would benefit from additional detail to make the proof verifiable. In the revised manuscript, we will expand the Theoretical analysis section to include a complete derivation using the stochastic block model as the generative model. We will provide the closed-form expression for the mutual information gain and analyze the conditions under which negative edges provide significant information. Additionally, we will include a discussion on the sensitivity to deviations from the block-diagonal structure, supported by both theoretical bounds and empirical checks on perturbed graphs. These additions will directly address the verifiability of the central inequality. revision: yes
Referee: Experimental section: the abstract asserts consistent outperformance “over simulated and real datasets in varying data regimes” yet reports neither error bars, number of runs, nor basic dataset statistics (node/edge counts, label rates, homophily coefficients). This prevents assessment of whether the reported gains are statistically reliable or confined to the exact regimes where the theory predicts benefit.

Authors: We acknowledge the omission of these important experimental details. In the revised version, we will augment the Experimental section with error bars computed over multiple independent runs (specifically, we will report means and standard deviations over 10 runs for each setting). We will also include a table summarizing key dataset statistics, including the number of nodes, edges, label rates, and homophily coefficients for both simulated and real-world graphs. This will enable readers to better evaluate the statistical reliability and the alignment with the theoretical regimes. revision: yes
Referee: § on CMP architecture: the soft positive-semidefinite constraint is motivated by the information-gain result, but the manuscript does not show how the constraint is enforced during training (projection, regularization term, or parameterization) nor whether it remains effective when homophily or density assumptions are only approximately satisfied.

Authors: Thank you for highlighting this gap in the presentation. The soft positive-semidefinite constraint is enforced through a regularization term in the training objective, where we add a penalty proportional to the sum of the negative eigenvalues of the weight matrix to encourage positive semidefiniteness. We will include the precise mathematical formulation and implementation details in the revised CMP architecture section. To address the effectiveness under approximate assumptions, we will add new experiments varying the homophily coefficient and edge density around the high-homophily, high-density regime, demonstrating that the performance gains persist even when the assumptions hold only approximately. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theoretical proof and CMP design are self-contained.

full rationale

The abstract states a theoretical analysis proving information gain from negative edges under low label rates, high homophily, and high edge density, then introduces CMP motivated by that insight with soft PSD constraints. No equations, fitted parameters, or self-citations are exhibited that would reduce the claimed gains or architecture to inputs by construction. The derivation chain is independent: the proof supplies external justification for the contrastive updates rather than redefining or fitting the target result. Empirical outperformance is presented as validation across regimes, not a statistically forced prediction. This is the expected honest non-finding for a paper whose central claims rest on stated assumptions and new architecture rather than tautological reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that negative edges supply information gain under the stated regimes; no free parameters or invented entities beyond the new CMP architecture itself are described in the abstract.

axioms (1)

domain assumption Negative edges provide significant information gain in low-label-rate, high-homophily, high-edge-density regimes.
This premise is invoked to motivate both the theoretical analysis and the design of CMP.

invented entities (1)

Contrastive Message Passing (CMP) architecture no independent evidence
purpose: General message-passing layer that reasons over positive and negative edges using soft positive-semidefinite constraints on weights.
New proposed component whose independent evidence is limited to the experiments summarized in the abstract.

pith-pipeline@v0.9.0 · 5701 in / 1317 out tokens · 46756 ms · 2026-05-20T12:27:16.662131+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1 ... IG_pos(s,r,d) = n r d ΔH+(s,d) f_pos ... IG_neg ... R_neg ... (Corollary 2.2)
IndisputableMonolith/Foundation/AlphaCoordinateFixation alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Soft-PSD(W) = Q Λ̂ Q⁻¹ ... τ = σ(±c(h_i,h_j)(1+β)) ... W₊,W₋ ⪰ 0 (Eq. 4-5)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Bhagat, G

S. Bhagat, G. Cormode, and S. Muthukrishnan. Node classification in social networks.Social network data analytics, pages 115–148, 2011

work page 2011

[2] [2]

S. Chen, J. Chen, S. Zhou, B. Wang, S. Han, C. Su, Y . Yuan, and C. Wang. Sigformer: Sign- aware graph transformer for recommendation. InProceedings of the 47th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 1274–1284, 2024

work page 2024

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work page arXiv 1905

[4] [4]

Chopra, R

S. Chopra, R. Hadsell, and Y . LeCun. Learning a similarity metric discriminatively, with application to face verification. In2005 IEEE computer society conference on computer vision and pattern recognition (CVPR’05), volume 1, pages 539–546. IEEE, 2005

work page 2005

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Chuang, J

C.-Y . Chuang, J. Robinson, Y .-C. Lin, A. Torralba, and S. Jegelka. Debiased contrastive learning. Advances in neural information processing systems, 33:8765–8775, 2020

work page 2020

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M. Du Plessis, G. Niu, and M. Sugiyama. Convex formulation for learning from positive and unlabeled data. InInternational conference on machine learning, pages 1386–1394. PMLR, 2015

work page 2015

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C. Elkan and K. Noto. Learning classifiers from only positive and unlabeled data. InProceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 213–220, 2008

work page 2008

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A fair comparison of graph neural networks for graph classification

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Kiryo, G

R. Kiryo, G. Niu, M. C. Du Plessis, and M. Sugiyama. Positive-unlabeled learning with non-negative risk estimator.Advances in neural information processing systems, 30, 2017

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Kreuzer, D

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Z. Liu, C. Wang, J. Xu, C. Wu, K. Zheng, Y . Song, N. Mou, and K. Gai. Pane-gnn: Unifying positive and negative edges in graph neural networks for recommendation.arXiv preprint arXiv:2306.04095, 2023

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Prakash, D

A. Prakash, D. Bermperidis, and S. Chennu. Evaluating performance and bias of negative sampling in large-scale sequential recommendation models.arXiv preprint arXiv:2410.17276, 2024

work page arXiv 2024

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Contrastive learning with hard negative samples,

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