REVIEW 3 major objections 5 minor 19 references
Graph Pooling by Local Cluster Selection
T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read Graph pooling by local cluster selection yields sparse, adaptive, edge-building coarsening.
desk verdict LCPool offers a clean local-assignment framework and a usable sparse pooling operator, but its own experimental table contradicts the headline claim that it ranks highest on every backbone. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the local cluster selection identity: when the assignment matrix $S$ is replaced by the adjacency-with-self-loops $A^* = I_N + A$, the pooled adjacency for the selected index set $\hat{i}$ becomes $A' = \mathrm{ones}(A + A^T A + A^2 + A^T A^2)(\hat{i},\hat{i})$, which for undirected graphs collapses to $A' = \mathrm{ones}(A + A^2 + A^3)(\hat{i},\hat{i})$. The identity does the main work because it converts a learned dense assignment into a sparse, parameter-free edge update based on one-hop and two-hop walks, and it is what makes adaptive graph sizes and sparse matrix multiplication possible. The second mechanism is the LCSMP score function $h_i = L_s(L_{fd}(\sum_{k\in N(i)} L_d(\tilde{x}_i - \tilde{x}_k)) + L_x(\tilde{x}_i))$, which learns to detect clusters from differences among local neighbours rather than from node features alone.
What would settle it
Re-run the four backbones on the same six datasets with multiple random splits and paired significance tests; if LCPool's average rank no longer beats NoPool and SAGPool, then the competitiveness claim is not supported.
Extended reading notes
Core claim
In the paper's own framing, the central claim is that a locally supported assignment matrix $S\in\mathbb{R}^{|V|\times|V|}$, whose entries are non-zero only where the adjacency matrix with self-loops $A^* = I_N + A$ is non-zero, is enough to generate pooled graphs with the advertised advantages. From this starting point the paper proves that the pooled adjacency matrix can be written as $A' = \mathrm{ones}(S'^T A S')$, and that under the common conditions of an unweighted graph and a one-hop feature-update function the assignment matrix can be replaced by $I_N + A$, giving $A' = \mathrm{ones}(A + A^T A + A^2 + A^T A^2)(\hat{i},\hat{i})$ and, for undirected graphs, $A' = \mathrm{ones}(A + A^2 + A^3)(\hat{i},\hat{i})$. The paper also argues that ordinary node selection is a special case of this framework with $S = I_N$, and that the separate cluster-function layer can be dropped when the preceding convolution already satisfies the one-hop condition. Its proposed LCPool operator selects the top-scored clusters using a learned score built from transformed neighbour differences, and the experimental section claims that this operator is more competitive and expressive than existing node-selection and dense-assignment pooling approaches.
Load-bearing premise
The claim that LCPool is the most competitive pooling operator rests on treating the average accuracy over ten runs on one fixed 8:1:1 split per dataset, with hyperparameters tuned only until every model converges, as a reliable comparison.
Editorial extensions
If this is right
- Pooled graphs keep a size proportional to the input graph, so small graphs are not forced upward and large graphs are not excessively shrunk, unlike dense-assignment pooling with a fixed second dimension.
- Because the assignment matrix is local and sparse, LCPool can be implemented with sparse matrix multiplications and therefore interoperates with sparse graph convolution layers.
- Edges are constructed between selected clusters whenever their contributors are connected, and original edges are preserved when the diagonal of $S$ is non-zero, so the coarsened graph is more connected than a pure node-selection output.
- For unweighted undirected graphs, the pooled adjacency is obtained directly as $\mathrm{ones}(A + A^2 + A^3)(\hat{i},\hat{i})$, so no assignment matrix needs to be formed or stored during pooling.
- The operator can be inserted into standard GNN backbones for graph classification, and the paper reports that it achieves the highest accuracy in the largest number of backbone-task combinations and the best average rank across the compared pooling methods.
Reading between the lines
- If the adjacency identity is taken as a definition of a local cluster, then pooling is effectively selecting nodes whose two-hop and three-hop walk patterns are dense; this suggests a testable extension in which the polynomial degree $r$ in $\mathrm{ones}(A + \cdots + A^r)$ is varied and accuracy is tracked on graphs with different diameter distributions.
- The Laplacian-difference score connects LCPool to spectral intuition, but the paper does not analyze it spectrally; one could test whether the learned score correlates with low-frequency eigenvectors, which would give an interpretation of why cluster selection helps.
- The experimental claim is only as strong as the fixed-split, ten-run protocol; a natural robustness check is repeated cross-validation with paired significance tests across multiple splits, which the paper does not report.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a graph pooling framework, local assignment selection, in which an assignment matrix S is constrained to have support only on one-hop neighborhoods; it then specializes to local cluster selection, where S can be replaced by I+A when edge weights are ignored, giving the closed-form edge update A' = ones(A + A^2 + A^3)(i_hat,i_hat) for undirected graphs. The proposed operator, LCPool, uses a new score function LCSMP that aggregates transformed feature differences among local neighbors, and is evaluated on six TUDatasets across four GNN backbones against five pooling baselines. The paper claims that LCPool is more competitive and expressive than the baselines, that it ranks highest for every backbone, and that the variant lcpool* with an extra GCN layer as v(·) does not outperform LCPool, thereby validating Proposition 7.
Significance. The theoretical framework is a useful formalization: it identifies a middle ground between node selection and dense assignment, highlights sparsity and adaptivity, and provides a simple, sparse-friendly edge update formula. Proposition 3 correctly embeds node selection as a special case, and Proposition 6 gives a compact derivation of the pooled adjacency. LCSMP is a reasonable and interpretable scoring layer. However, the empirical support as printed is contradicted by the paper's own tables, so the claimed practical advantage is not established. If the experiments are revised and the claims are corrected, the framework could be a modest but publishable contribution to graph pooling methodology.
major comments (3)
- [Section 4.5, Table 6] The statement that LCPool 'ranks highest for every backbone' is contradicted by Table 6: on GNNc_h, lcpool has average rank 3.5, worse than nopool's 2.33, and on GNNc_p it only ties sagpool at 3.17. Since these ranks are derived from Tables 2-5, the discrepancy is internal and does not depend on any contested baseline choice. The ranking claim must be revised, or the ranking methodology must be clarified (for example, by excluding nopool, by using per-task wins, or by accounting for statistical ties).
- [Section 4.5, Tables 2-5] The claim that the variant lcpool* 'does not outperform' LCPool is not supported by the reported numbers. On NCI1, lcpool* is higher in all four backbones (e.g., 81.80 vs 79.10 on GNNh, 81.00 vs 78.61 on GNNc_h, 81.80 vs 79.49 on GNNp, and 81.65 vs 80.61 on GNNc_p); on Mutagenicity, lcpool* is higher on GNNc_h (81.61 vs 80.51) and GNNc_p (80.83 vs 80.48); on DD, lcpool* is higher on GNNp (75.08 vs 74.32). Thus, as written, the comparison does not validate Proposition 7. The authors should either report significance tests and a clearly defined aggregate criterion, or explicitly acknowledge that lcpool* outperforms lcpool on several datasets and discuss the confound of additional model capacity.
- [Section 4.5 and Appendix B] The experimental protocol rests on one fixed 8:1:1 split, ten runs, and no paired significance tests or multiple splits. Many reported differences are within one standard deviation (e.g., Table 3, PROTEINS: lcpool 74.91±1.89 vs nopool 74.02±1.67; Table 5, ENZYMES: lcpool 71.83±6.69 vs diffpool 76.17±3.88). Consequently, even after correcting the ranking claim, the paper would not establish that LCPool is systematically more competitive or expressive. Paired tests over multiple splits are needed to support the central empirical conclusion.
minor comments (5)
- [Throughout] There are numerous typos and grammatical errors, including 'shrinked', 'recommander', 'stuctures', 'adpoting', 'exsits', 'trainning', 'acer', and 'vaild'; these should be corrected in a revision.
- [Section 3.1, Prerequisite 1] The condition 'sij is non-zero only when aij is not zero' should refer to entries of A* = I_N + A, not the original adjacency A; otherwise the requirement that a node contributes to itself is inconsistent with aii = 0 in the original adjacency matrix.
- [Section 3.3.2] The sentence 'We assumpt that the input and coarsened graphs are undirected and have unweighted edges' should be reworded as 'We assume...'.
- [Table 2] The dataset name 'Muatagenicity' is misspelled; it should be 'Mutagenicity'.
- [Section 5] The conclusion repeats the empirical claims contradicted by Table 6 and the lcpool* comparison; it should be updated to match the revised results.
Circularity Check
No circular derivation: the simplified edge formula follows from the stated support assumption, and the proposed operator is validated on external benchmark accuracies.
full rationale
The paper's derivation chain is self-contained. Local cluster selection assumes S has the same support as I+A (Prerequisites 1 and 4) and unweighted edges, so Lemma 5's replacement of S by I+A in A'=S^T A S preserves non-zero positions by construction; this is a direct algebraic simplification, not a fitted prediction. Proposition 7 (dismissing v) is justified by comparing two trained variants (LCPool vs. LCPool*) on external graph-classification benchmarks, which is an empirical check rather than a derivation from the conclusion. The claimed 'ranks highest for every backbone' is contradicted by the paper's own Table 6 (e.g., GNNc_h: nopool 2.33 vs lcpool 3.5), but that is an internal consistency/correctness problem, not circularity. No load-bearing self-citations or imported uniqueness theorems appear; references to prior frameworks (Grattarola et al. [6], ASAPool [15]) are contextual. Therefore no step reduces to its own input.
Assumptions & free parameters
free parameters (3)
- Pooling ratio k in topk(h) =
not stated
- LCSMP hidden dimensions =
not stated
- Training hyperparameters =
lr=0.0005, hidden=128, batch=32, patience=50, max epochs=500, split 8:1:1
assumptions (4)
- standard math Message passing formalization of graph convolution and the SRC selection-reduction-connection characterization of pooling
- domain assumption For local cluster selection, input and coarsened graphs are undirected and have unweighted edges
- domain assumption The assignment matrix support is exactly the one-hop neighborhood including self-loops, so S can be replaced by I+A for edge construction
- ad hoc to paper The cluster function v(·) can be dismissed when the preceding convolution layer already satisfies Lemma 5
Cite this review
Pith. "Pith review of Graph Pooling by Local Cluster Selection." pith.science (2026). https://pith.science/paper/Z3FOSSFK
@misc{pith2026241116615,
author = {Pith},
title = {Pith review of: Graph Pooling by Local Cluster Selection},
year = {2026},
howpublished = {\url{https://pith.science/paper/Z3FOSSFK}},
note = {Machine review of arXiv:2411.16615}
}
read the original abstract
Graph pooling is a family of operations which take graphs as input and produce shrinked graphs as output. Modern graph pooling methods are trainable and, in general inserted in Graph Neural Networks (GNNs) architectures as graph shrinking operators along the (deep) processing pipeline. This work proposes a novel procedure for pooling graphs, along with a node-centred graph pooling operator.
Figures
Figures from the paper (1 more)
Reference graph
Works this paper leans on
-
[1]
A comprehensive survey on deep graph representation learning
Wei Ju, Zheng Fang, Yiyang Gu, Zequn Liu, Qingqing Long, Ziyue Qiao, Yifang Qin, Jianhao Shen, Fang Sun, Zhiping Xiao, Junwei Yang, Jingyang Yuan, Yusheng Zhao, Yifan Wang, Xiao Luo, and Ming Zhang. A comprehensive survey on deep graph representation learning. Neural Networks, 173:106207, May 2024. ISSN 0893-6080. doi: 10.1016/j.neunet.2024.106207. URL ht...
arXiv 2024
-
[3]
Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S
Zhenqin Wu, Bharath Ramsundar, Evan N. Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S. Pappu, Karl Leswing, and Vijay S. Pande. Moleculenet: A benchmark for molecular machine learning. CoRR, abs/1703.00564, 2017. URL http://arxiv.org/abs/1703.00564. 1
arXiv 2017
-
[4]
Multivariate time series imputation by graph neu- ral networks
Andrea Cini, Ivan Marisca, and Cesare Alippi. Multivariate time series imputation by graph neu- ral networks. CoRR, abs/2108.00298, 2021. URL https://arxiv.org/abs/2108.00298. 1
arXiv 2021
-
[5]
Graph pooling for graph neural networks: Progress, challenges, and opportunities, 2023
Chuang Liu, Yibing Zhan, Jia Wu, Chang Li, Bo Du, Wenbin Hu, Tongliang Liu, and Dacheng Tao. Graph pooling for graph neural networks: Progress, challenges, and opportunities, 2023. URL https://arxiv.org/abs/2204.07321. 1
arXiv 2023
-
[6]
Understanding pooling in graph neural networks
Daniele Grattarola, Daniele Zambon, Filippo Maria Bianchi, and Cesare Alippi. Understanding pooling in graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 35(2):2708–2718, February 2024. ISSN 2162-2388. doi: 10.1109/tnnls.2022.3190922. URL http://dx.doi.org/10.1109/TNNLS.2022.3190922. 1, 2
arXiv 2024
-
[7]
Rex Ying, Jiaxuan You, Christopher Morris, Xiang Ren, William L. Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling, 2019. URL https://arxiv.org/abs/1806.08804. 1, 2, 7
arXiv 2019
-
[8]
Diego P. P. Mesquita, Amauri H. Souza Jr., and Samuel Kaski. Rethinking pooling in graph neu- ral networks. CoRR, abs/2010.11418, 2020. URL https://arxiv.org/abs/2010.11418. 1 9 LCPool
arXiv 2010
-
[9]
Hierarchical Representation Learning in Graph Neural Networks with Node Decimation Pooling
Filippo Maria Bianchi, Daniele Grattarola, Lorenzo Livi, and Cesare Alippi. Hierarchi- cal representation learning in graph neural networks with node decimation pooling. CoRR, abs/1910.11436, 2019. URL http://arxiv.org/abs/1910.11436. 1
work page Pith review arXiv 1910
Show all 19 references
-
[10]
A non-negative factorization approach to node pooling in graph convolutional neural networks, 2019
Davide Bacciu and Luigi Di Sotto. A non-negative factorization approach to node pooling in graph convolutional neural networks, 2019. URL https://arxiv.org/abs/1909.03287. 1
2019 arXiv
-
[11]
Dhillon, Yuqiang Guan, and Brian Kulis
Inderjit S. Dhillon, Yuqiang Guan, and Brian Kulis. Weighted graph cuts without eigenvectors a multilevel approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (11):1944–1957, 2007. doi: 10.1109/TPAMI.2007.1115. 1
1944
-
[12]
Graph u-nets
Hongyang Gao and Shuiwang Ji. Graph u-nets. CoRR, abs/1905.05178, 2019. URL http: //arxiv.org/abs/1905.05178. 1, 2, 7
1905 arXiv
-
[13]
Self-attention graph pooling
Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors,Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 3734–
-
[14]
Mincut pooling in graph neural networks
Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Mincut pooling in graph neural networks. CoRR, abs/1907.00481, 2019. URL http://arxiv.org/abs/1907.00481. 1, 2, 7
1907 arXiv
-
[15]
Asap: Adaptive structure aware pooling for learning hierarchical graph representations, 2020
Ekagra Ranjan, Soumya Sanyal, and Partha Pratim Talukdar. Asap: Adaptive structure aware pooling for learning hierarchical graph representations, 2020. URL https://arxiv.org/ abs/1911.07979. 1, 3, 7
2020 arXiv
-
[16]
A comprehensive graph pooling benchmark: Effectiveness, robustness and generalizability, 2024
Pengyun Wang, Junyu Luo, Yanxin Shen, Siyu Heng, and Xiao Luo. A comprehensive graph pooling benchmark: Effectiveness, robustness and generalizability, 2024. URL https: //arxiv.org/abs/2406.09031. 1
2024
-
[17]
Schoenholz, Patrick F
Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. CoRR, abs/1704.01212, 2017. URL http: //arxiv.org/abs/1704.01212. 1
2017 arXiv
-
[18]
Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neumann
Christopher Morris, Nils M. Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neumann. Tudataset: A collection of benchmark datasets for learning with graphs. In ICML 2020 Workshop on Graph Representation Learning and Beyond (GRL+ 2020) , 2020. URL www.graphlea...
2020
-
[2019]
URL http://arxiv.org/abs/1912.11730. 1
1912 arXiv
-
[3743]
URL https://proceedings.mlr.press/v97/lee19c
PMLR, 09–15 Jun 2019. URL https://proceedings.mlr.press/v97/lee19c. html. 1, 2, 6, 7
2019
Reviewed August 12, 2026 · model on record in the stance chip above.
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