Adaptive Recurrent Message Passing for Test Time Computing on Graphs

Junshu Sun; Qingming Huang; Shuhui Wang; Wanxing Chang

arxiv: 2606.22462 · v1 · pith:PFNYKY4Unew · submitted 2026-06-21 · 💻 cs.LG

Adaptive Recurrent Message Passing for Test Time Computing on Graphs

Junshu Sun , Wanxing Chang , Qingming Huang , Shuhui Wang This is my paper

Pith reviewed 2026-06-26 10:47 UTC · model grok-4.3

classification 💻 cs.LG

keywords recurrent graph modelstest-time computingadaptive convergencemessage passinggraph neural networksparameter-free adaptation

0 comments

The pith

Step dependence is necessary and sufficient for adaptively convergent recurrent processes on graphs, enabling parameter-free test-time computing via AdaR.

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

The paper derives step dependence as the necessary and sufficient condition for a recurrent message-passing process on graphs to converge adaptively. It introduces AdaR, which encodes normalized step information and representation-target relations into the updates while using gradient-based supervision to steer the recurrence. This design supports test-time adaptation across downstream tasks without any parameter changes. A sympathetic reader would see the result as removing the fixed-architecture barrier that currently separates pre-trained graph models from the flexible test-time behavior seen in other domains. Experiments confirm outperformance over strong baselines in both inductive and transductive regimes.

Core claim

We rigorously derive step dependence as a necessary and sufficient condition for an adaptively convergent recurrent process. Building on this foundation, we propose AdaR, an Adaptive Recurrent graph model, empowering flexible test-time computing on various downstream tasks without changing model parameters. To enable adaptive inference, AdaR explicitly encodes normalized step information and representation-target relations into the recurrent updates. To ensure convergence of the recurrent process, AdaR employs gradient-based supervision signals that guide representation updates throughout the recurrence.

What carries the argument

Step dependence, the condition that makes an adaptively convergent recurrent process possible on graphs by tying update behavior to the current computation step.

If this is right

AdaR performs test-time computing on graphs without altering its pre-trained parameters.
The same model can be applied to both inductive and transductive downstream tasks.
Recurrent message passing becomes viable for foundation-model-style generalization once step dependence is enforced.
Gradient supervision during recurrence guarantees convergence without task-specific retraining.

Where Pith is reading between the lines

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

Variable numbers of recurrence steps could be chosen at inference time to trade compute for accuracy on new graphs.
The same step-dependence principle may transfer to recurrent architectures outside the graph domain.

Load-bearing premise

The recurrent process can be made to converge reliably by encoding normalized step information and representation-target relations together with gradient-based supervision signals.

What would settle it

Training or running AdaR without the normalized step-information encoding and observing whether the recurrent process still converges and adapts on standard graph benchmarks.

Figures

Figures reproduced from arXiv: 2606.22462 by Junshu Sun, Qingming Huang, Shuhui Wang, Wanxing Chang.

**Figure 1.** Figure 1: Diversity of Optimal Layer Configurations Across Datasets. Graph models exhibit varying optimal numbers of layers or steps across different datasets. This indicates that effective feature extraction requires the models to adjust their receptive field based on specific datasets. In consequence, pre-trained graph models with a fixed number of layers lack the flexibility to be adapted to diverse datasets. dow… view at source ↗

**Figure 2.** Figure 2: Comparison Between Traditional and Recurrent Graph Model. (a) Traditional models stack multiple cascade layers. (b) In contrast, recurrent models employ a recurrent layer, which can be iteratively reused to adapt to different receptive field requirements. for downstream tasks. Specifically, some methods transfer external knowledge from LMs to graph models via strategies such as knowledge distillation (Xia… view at source ↗

**Figure 3.** Figure 3: Adaptability of AdaR. To enable a target-aware recurrent process, AdaR introduces relations with targets to compute representation updates. However, the number of targets varies across tasks. To tackle this problem, a proxy-based relation computing strategy is designed, where a fixed number of pseudo nodes aggregate information from different numbers of targets, serving as target proxies for relation compu… view at source ↗

**Figure 4.** Figure 4: Test-time Computing under Different Design Choices. (a)-(c) studies the effectiveness of different encoded information, where w/o. S denotes no access to step information, S(∆t = 1) denotes encoding step information with a fixed unit step, w/o. R denotes not employing relations with targets. (d) presents the results of our AdaR. (e)-(f) studies the effectiveness of the supervision signals, where w/o. Lfull… view at source ↗

**Figure 5.** Figure 5: Iteration Exiting Strategy. 0.6 0.8 1.0 1.2 1.4 1.6 Average Inference Time (s) 0 10 20 30 40 50 Model Performance Training Time (s) GCNII N 2 AdaR [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Accurcay-Computing Curve. Bubble size denotes training time. ‘x’ denotes out-of-memory configurations. Effectiveness under Constrained Resources. Different downstream tasks require various optimal receptive fields. To satisfy such requirements, graph nodes must be able to reach nodes at any distance via message passing. Previous works, such as GCNII (Chen et al., 2020), directly achieve this by implementi… view at source ↗

read the original abstract

Pre-trained foundation models have demonstrated remarkable success in many domains, enabling a unified backbone to generalize across diverse downstream tasks. However, extending this paradigm to graph learning remains challenging due to the intrinsic mismatch between graph data and fixed architectural designs. In this work, we show that this limitation can be overcome via recurrent graph models. To achieve this, we conduct a systematic theoretical analysis, rigorously deriving step dependence as a necessary and sufficient condition for an adaptively convergent recurrent process. Building on this foundation, we propose AdaR, an Adaptive Recurrent graph model, empowering flexible test-time computing on various downstream tasks without changing model parameters. To enable adaptive inference, AdaR explicitly encodes normalized step information and representation-target relations into the recurrent updates. To ensure convergence of the recurrent process, AdaR employs gradient-based supervision signals that guide representation updates throughout the recurrence. Empirical results demonstrate that AdaR consistently outperforms strong baselines in both inductive and transductive settings.

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 abstract asserts a derivation that step dependence is necessary and sufficient for adaptive convergence but supplies no definitions, theorem, or proof sketch, so the central claim cannot be checked.

read the letter

The abstract states that a systematic theoretical analysis derives step dependence as necessary and sufficient for an adaptively convergent recurrent process on graphs, then builds AdaR on that to enable parameter-free test-time computing. That is the main claim a reader needs to know. Without the actual definitions of step dependence, the formal statement, assumptions, or any proof elements, it is impossible to assess whether the result is correct or reduces to something already known.

The work targets a genuine limitation: pre-trained graph models struggle with task diversity because architectures are fixed at training time. AdaR tries to address this by making the recurrent updates depend on normalized step information and representation-target relations, with gradient-based signals to drive convergence. If the full paper shows that these choices produce reliable adaptation across inductive and transductive settings, that would be a concrete engineering contribution for graph ML.

The soft spots are exactly where the stress-test note flags them. The necessity and sufficiency claim is presented at the level of design intent only. No equations appear in the abstract, so there is no way to verify internal consistency or independence from fitted parameters. The empirical section asserts consistent outperformance over strong baselines but gives no dataset names, no error bars, no ablation results, and no details on how convergence was measured. These omissions make it hard to judge whether the reported gains are robust.

This paper is aimed at researchers already working on recurrent GNN variants and test-time adaptation. Someone in that sub-area could extract useful design patterns from the encoding choices even if the theory needs tightening. It is not yet clear whether the result is new relative to earlier recurrent message-passing work, because the abstract does not engage the prior literature in enough detail.

I would send the manuscript to peer review rather than desk reject. The topic is relevant inside graph ML, and a referee can check whether the full derivations and experiments hold up. If the theory section turns out to be thin, that can be addressed in revision.

Referee Report

2 major / 1 minor

Summary. The paper claims that a systematic theoretical analysis derives step dependence as a necessary and sufficient condition for an adaptively convergent recurrent process on graphs. It proposes AdaR, an adaptive recurrent graph model that encodes normalized step information and representation-target relations into updates and uses gradient-based supervision to ensure convergence, thereby enabling parameter-free test-time computing; empirical results are said to show consistent outperformance over baselines in inductive and transductive settings.

Significance. If the derivation of necessity and sufficiency holds and the empirical results prove robust, the work could establish a principled mechanism for adaptive recurrence in graph models, addressing architectural mismatch with graph data via test-time adaptation without parameter changes. The focus on a formal condition for convergence would be a notable strength if rigorously established.

major comments (2)

[Abstract] Abstract: the central claim that step dependence is necessary and sufficient for an adaptively convergent recurrent process is asserted without any definition of 'step dependence', formal theorem statement, assumptions, or proof sketch, rendering the derivation unevaluable for correctness or internal consistency.
[Abstract] Abstract (AdaR design paragraph): the convergence guarantee is described only via design elements (normalized step encoding, representation-target relations, gradient supervision) with no equations, analysis, or conditions showing how these elements produce reliable convergence or exploit the claimed step-dependence property.

minor comments (1)

[Abstract] Abstract lacks any mention of datasets, error bars, ablation controls, or statistical significance for the empirical superiority claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for highlighting issues with the clarity of our abstract. We agree that the abstract's brevity makes the central theoretical claim difficult to evaluate without additional context, and we will revise it to address both points while preserving its summary nature. The full definitions, theorems, equations, and proofs appear in Sections 3 and 4 of the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that step dependence is necessary and sufficient for an adaptively convergent recurrent process is asserted without any definition of 'step dependence', formal theorem statement, assumptions, or proof sketch, rendering the derivation unevaluable for correctness or internal consistency.

Authors: We agree the abstract does not define 'step dependence' or include a theorem statement. 'Step dependence' is defined in Section 3 as the explicit encoding of normalized iteration count into the recurrent update function. Theorem 1 states that step dependence is necessary and sufficient for adaptive convergence of the recurrent process on graphs, under the assumptions of Lipschitz continuity of the update and bounded graph degree. The proof sketch (contraction mapping for sufficiency; counterexample construction for necessity) is also in Section 3. We will revise the abstract to include a one-sentence definition and reference to Theorem 1. revision: yes
Referee: [Abstract] Abstract (AdaR design paragraph): the convergence guarantee is described only via design elements (normalized step encoding, representation-target relations, gradient supervision) with no equations, analysis, or conditions showing how these elements produce reliable convergence or exploit the claimed step-dependence property.

Authors: We agree the abstract summarizes design elements without equations or explicit conditions. The normalized step encoding appears in Equation (4), the target-relation term in Equation (5), and gradient supervision in the loss of Equation (7). Section 4.2 analyzes how these components enforce the step-dependent contraction required by Theorem 1, yielding convergence to a fixed point independent of initialization. We will revise the abstract to add a brief clause stating the convergence condition derived from step dependence. revision: yes

Circularity Check

0 steps flagged

No circularity detected; asserted theoretical derivation not provided for inspection

full rationale

The paper asserts that a systematic theoretical analysis was conducted to derive step dependence as a necessary and sufficient condition for an adaptively convergent recurrent process, upon which AdaR is built. However, the provided text contains no equations, no formal definition of step dependence, no theorem statement, no assumptions, and no proof. The AdaR design is described only at the level of intent (encoding normalized step information, representation-target relations, and gradient supervision). No self-definitional, fitted-input, or self-citation patterns can be exhibited because the derivation chain itself is not visible. The central claim therefore cannot be reduced to its inputs by construction from the given material, making the derivation self-contained against external benchmarks by default.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on an unshown theoretical derivation of step dependence and on the assumption that gradient supervision will produce convergent behavior in practice; no free parameters, invented entities, or additional axioms are visible in the abstract.

axioms (1)

domain assumption Step dependence is necessary and sufficient for adaptive convergence of recurrent graph message passing
Stated as the outcome of the systematic theoretical analysis

pith-pipeline@v0.9.1-grok · 5690 in / 1048 out tokens · 22759 ms · 2026-06-26T10:47:51.858234+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 1 canonical work pages

[1]

Hu, Z., Li, Y ., Chen, Z., Wang, J., Liu, H., Lee, K., and Ding, K

Association for Computing Machinery. Hu, Z., Li, Y ., Chen, Z., Wang, J., Liu, H., Lee, K., and Ding, K. Let’s Ask GNN: Empowering Large Language Model for Graph In-Context Learning. In Al-Onaizan, Y ., Bansal, M., and Chen, Y .-N. (eds.),Findings of the Asso- ciation for Computational Linguistics: EMNLP 2024, pp. 1396–1409, Miami, Florida, USA,

2024
[2]

Can GNN be Good Adapter for LLMs? InProceedings of the ACM Web Conference 2024, pp

Huang, X., Han, K., Yang, Y ., Bao, D., Tao, Q., Chai, Z., and Zhu, Q. Can GNN be Good Adapter for LLMs? InProceedings of the ACM Web Conference 2024, pp. 893–904, New York, NY , USA,

2024
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X., and Li, J

Li, Y ., Wang, P., Li, Z., Yu, J. X., and Li, J. ZeroG: Investi- gating Cross-dataset Zero-shot Transferability in Graphs. InACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1725–1735, New York, NY , USA, 2024b. Association for Computing Machinery. Liu, H., Feng, J., Kong, L., Liang, N., Tao, D., Chen, Y ., and Zhang, M. One For All: Towa...

2023
[4]

Scarselli, F., Gori, M., Tsoi, A

ISSN 1941-0476. Scarselli, F., Gori, M., Tsoi, A. C., Hagenbuchner, M., and Monfardini, G. The Graph Neural Network Model.IEEE Transactions on Neural Networks, 20(1):61–80,

1941
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Pitfalls of Graph Neural Network Evaluation

Shchur, O., Mumme, M., Bojchevski, A., and G¨unnemann, S. Pitfalls of Graph Neural Network Evaluation. arXiv:1811.05868 [cs, stat],

Pith/arXiv arXiv
[6]

Re- lieving the Over-Aggregating Effect in Graph Transform- ers

Sun, J., Chang, W., Yang, C., Huang, Q., and Wang, S. Re- lieving the Over-Aggregating Effect in Graph Transform- ers. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025a. Sun, L., Huang, Z., Zhou, S., Wan, Q., Peng, H., and Yu, P. S. RiemannGFM: Learning a Graph Foundation Model from Structural Geometry. InTHE WEB CONFERE...

2025
[7]

Xia, L., Kao, B., and Huang, C

doi: 10.52202/079017-3412. Xia, L., Kao, B., and Huang, C. OpenGraph: Towards Open Graph Foundation Models. In Al-Onaizan, Y ., Bansal, M., and Chen, Y .-N. (eds.),Findings of the Association for Computational Linguistics: EMNLP 2024, pp. 2365– 2379, Miami, Florida, USA,

work page doi:10.52202/079017-3412 2024
[8]

Yu, X., Fang, Y ., Liu, Z., Wu, Y ., Wen, Z., Bo, J., Zhang, X., and Hoi, S. C. H. A Survey of Few-Shot Learning on Graphs: From Meta-Learning to Pre-Training and Prompt Learning, 2024a. Yu, X., Zhou, C., Fang, Y ., and Zhang, X. MultiGPrompt for Multi-Task Pre-Training and Prompting on Graphs. InProceedings of the ACM Web Conference 2024, pp. 515–526, Ne...

2024
[9]

(A14) By uniform approximation, the Lipschitz condition, and the triangle inequality, we obtain ∥es+1∥ ≤(1 +L H∆t)∥es∥+ε∆t+K 1∆t2,(A15) for some constantK 1 >0. Applying induction and usinge 0 = 0, we have ∥es+1∥ ≤(1 +L H∆t)s+1∥e0∥|{z} =0 + sX i=0 (1 +L H∆t)i(ε∆t+K 1∆t2) = (1 +L H∆t)s −1 LH∆t (ε∆t+K 1∆t2) = exp[sln(1 +L H∆t)]−1 LH (ε+K 1∆t) ≤ exp[sLH∆t]−1...

2024
[10]

Regarding implementations, pseudo nodes in AdaR are shared parameters for all graphs

or to enhance the cross-graph communication (Zhao et al., 2024), yet AdaR employs pseudo nodes to unify different targets. Regarding implementations, pseudo nodes in AdaR are shared parameters for all graphs. Conversely, pseudo nodes in previous works either cannot be jointly optimized with the overall framework or fail to transfer across tasks (Liu et al...

2024
[11]

In contrast, fixing the total time with T= 1 rescales ts to [0,1]

proportional to T and large embedding deviations. In contrast, fixing the total time with T= 1 rescales ts to [0,1] . Therefore, the maximum shift is bounded by the constant wk independent ofT, implying smaller embedding shifts. Empirical Analysis.To empirically validate our encoding strategy, we select the iteration budget from two distinct ranges, i.e.s...

2008
[12]

Specifically, CoauthorCS and CoauthorPhysics are derived from the Microsoft Academic Graph and model academic collaboration networks

and homophilic datasets (CoauthorCS, CoauthorPhysics, AmazonPhoto, and AmazonComputers) (Shchur et al., 2019). Specifically, CoauthorCS and CoauthorPhysics are derived from the Microsoft Academic Graph and model academic collaboration networks. AmazonComputers and AmazonPhoto are product co-purchase networks collected from Amazon. Questions is built from ...

2019
[13]

Models are selected based on the average performance on training datasets for the zero-shot setting, while based on validation sets for the transductive setting

For transductive experiments, the training epoch is set as 1,000 , the learning rate is set to 1×10 −3, the training iteration budget is selected from {2,4,8} , the test-time iteration budget is selected from {8,20,50,100,150,200,250,300} and the dropout probability is selected from {0,0.1,0.2,0.3} . Models are selected based on the average performance on...

2017

[1] [1]

Hu, Z., Li, Y ., Chen, Z., Wang, J., Liu, H., Lee, K., and Ding, K

Association for Computing Machinery. Hu, Z., Li, Y ., Chen, Z., Wang, J., Liu, H., Lee, K., and Ding, K. Let’s Ask GNN: Empowering Large Language Model for Graph In-Context Learning. In Al-Onaizan, Y ., Bansal, M., and Chen, Y .-N. (eds.),Findings of the Asso- ciation for Computational Linguistics: EMNLP 2024, pp. 1396–1409, Miami, Florida, USA,

2024

[2] [2]

Can GNN be Good Adapter for LLMs? InProceedings of the ACM Web Conference 2024, pp

Huang, X., Han, K., Yang, Y ., Bao, D., Tao, Q., Chai, Z., and Zhu, Q. Can GNN be Good Adapter for LLMs? InProceedings of the ACM Web Conference 2024, pp. 893–904, New York, NY , USA,

2024

[3] [3]

X., and Li, J

Li, Y ., Wang, P., Li, Z., Yu, J. X., and Li, J. ZeroG: Investi- gating Cross-dataset Zero-shot Transferability in Graphs. InACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1725–1735, New York, NY , USA, 2024b. Association for Computing Machinery. Liu, H., Feng, J., Kong, L., Liang, N., Tao, D., Chen, Y ., and Zhang, M. One For All: Towa...

2023

[4] [4]

Scarselli, F., Gori, M., Tsoi, A

ISSN 1941-0476. Scarselli, F., Gori, M., Tsoi, A. C., Hagenbuchner, M., and Monfardini, G. The Graph Neural Network Model.IEEE Transactions on Neural Networks, 20(1):61–80,

1941

[5] [5]

Pitfalls of Graph Neural Network Evaluation

Shchur, O., Mumme, M., Bojchevski, A., and G¨unnemann, S. Pitfalls of Graph Neural Network Evaluation. arXiv:1811.05868 [cs, stat],

Pith/arXiv arXiv

[6] [6]

Re- lieving the Over-Aggregating Effect in Graph Transform- ers

Sun, J., Chang, W., Yang, C., Huang, Q., and Wang, S. Re- lieving the Over-Aggregating Effect in Graph Transform- ers. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025a. Sun, L., Huang, Z., Zhou, S., Wan, Q., Peng, H., and Yu, P. S. RiemannGFM: Learning a Graph Foundation Model from Structural Geometry. InTHE WEB CONFERE...

2025

[7] [7]

Xia, L., Kao, B., and Huang, C

doi: 10.52202/079017-3412. Xia, L., Kao, B., and Huang, C. OpenGraph: Towards Open Graph Foundation Models. In Al-Onaizan, Y ., Bansal, M., and Chen, Y .-N. (eds.),Findings of the Association for Computational Linguistics: EMNLP 2024, pp. 2365– 2379, Miami, Florida, USA,

work page doi:10.52202/079017-3412 2024

[8] [8]

Yu, X., Fang, Y ., Liu, Z., Wu, Y ., Wen, Z., Bo, J., Zhang, X., and Hoi, S. C. H. A Survey of Few-Shot Learning on Graphs: From Meta-Learning to Pre-Training and Prompt Learning, 2024a. Yu, X., Zhou, C., Fang, Y ., and Zhang, X. MultiGPrompt for Multi-Task Pre-Training and Prompting on Graphs. InProceedings of the ACM Web Conference 2024, pp. 515–526, Ne...

2024

[9] [9]

(A14) By uniform approximation, the Lipschitz condition, and the triangle inequality, we obtain ∥es+1∥ ≤(1 +L H∆t)∥es∥+ε∆t+K 1∆t2,(A15) for some constantK 1 >0. Applying induction and usinge 0 = 0, we have ∥es+1∥ ≤(1 +L H∆t)s+1∥e0∥|{z} =0 + sX i=0 (1 +L H∆t)i(ε∆t+K 1∆t2) = (1 +L H∆t)s −1 LH∆t (ε∆t+K 1∆t2) = exp[sln(1 +L H∆t)]−1 LH (ε+K 1∆t) ≤ exp[sLH∆t]−1...

2024

[10] [10]

Regarding implementations, pseudo nodes in AdaR are shared parameters for all graphs

or to enhance the cross-graph communication (Zhao et al., 2024), yet AdaR employs pseudo nodes to unify different targets. Regarding implementations, pseudo nodes in AdaR are shared parameters for all graphs. Conversely, pseudo nodes in previous works either cannot be jointly optimized with the overall framework or fail to transfer across tasks (Liu et al...

2024

[11] [11]

In contrast, fixing the total time with T= 1 rescales ts to [0,1]

proportional to T and large embedding deviations. In contrast, fixing the total time with T= 1 rescales ts to [0,1] . Therefore, the maximum shift is bounded by the constant wk independent ofT, implying smaller embedding shifts. Empirical Analysis.To empirically validate our encoding strategy, we select the iteration budget from two distinct ranges, i.e.s...

2008

[12] [12]

Specifically, CoauthorCS and CoauthorPhysics are derived from the Microsoft Academic Graph and model academic collaboration networks

and homophilic datasets (CoauthorCS, CoauthorPhysics, AmazonPhoto, and AmazonComputers) (Shchur et al., 2019). Specifically, CoauthorCS and CoauthorPhysics are derived from the Microsoft Academic Graph and model academic collaboration networks. AmazonComputers and AmazonPhoto are product co-purchase networks collected from Amazon. Questions is built from ...

2019

[13] [13]

Models are selected based on the average performance on training datasets for the zero-shot setting, while based on validation sets for the transductive setting

For transductive experiments, the training epoch is set as 1,000 , the learning rate is set to 1×10 −3, the training iteration budget is selected from {2,4,8} , the test-time iteration budget is selected from {8,20,50,100,150,200,250,300} and the dropout probability is selected from {0,0.1,0.2,0.3} . Models are selected based on the average performance on...

2017