Node Embedding and Cosine Similarity for Efficient Maximum Common Subgraph Discovery

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper is dedicated to improving the McSplit algorithm for solving the Maximum Common Subgraph (MCS) problem by integrating heuristics based on Graph Neural Networks (GNNs). The topic is relevant, and the study combines the precision of a classical algorithm with the intelligence of modern machine learning approaches. The research is methodologically sound. The authors enhance McSplit by incorporating node embeddings, introduce new heuristics, and compare different selection strategies. A large number of graph pairs are considered in the experimental section. The results demonstrate a significant reduction in computation time and recursion depth while maintaining solution quality.

However, in order to improve the quality of the manuscript, I would like to draw the authors’ attention to the following comments and recommendations:

The literature review is rather narrow and lacks critical analysis of previous methods.

Although the results are presented in detail, the authors are encouraged to summarize the key indicators in a dedicated summary table to improve formal clarity and facilitate understanding.

The statement in the conclusions regarding “significant improvement” is not statistically supported. I recommend including at least a basic analysis of confidence or distribution to justify the claim.

The reference list requires revision. Several sources do not contain DOIs. In addition, reference [25] (Private Communication) cannot be considered a valid scientific reference in my opinion.

Furthermore, there is a substantial risk of self-plagiarism in this work, which should be clearly acknowledged and explained by the authors.

Author Response

We sincerely thank the editor and reviewers for their meticulous review and insightful suggestions. We have diligently addressed all comments in this revised version of the paper. We hope that the same Reviewers will review the paper and that our adjustments will satisfy them.

From a high-level perspective, our key revisions include:

• A comprehensive rectification of all major stylistic and English language issues, achieved through a careful revision and a pass with Grammarly Pro.
• A deeper analysis, from both the theoretical and experimental points of view, of the norm and cosine similarity approaches, their comparison, the NeuroMatch model, and the selection of the primary vital parameters (such as k).
• Additional experiments on challenging graph pairs to demonstrate the advantages of our proposed method more vividly. To underscore this, we have modified and incorporated new pie plots in Section 3.8, Figure 7, on Page 21, with a deeper description of the numerical advantages of our methods.
• A complete overhaul of the reference list, now including a more in-depth analysis of related papers. This action has expanded our reference count from 25 to 34.

The new version of the paper is approximately four pages longer than the original submission. We have highlighted all major modifications in blue within the paper for easy identification, intentionally omitting minor changes such as typos or small corrections.
The following sections provide explicit discussions of issues that required more than superficial corrections, presented as direct answers to the reviewers' comments.

***

**General Comment**
The paper is dedicated to improving the McSplit algorithm for solving the Maximum Common Subgraph (MCS) problem by integrating heuristics based on Graph Neural Networks (GNNs). The topic is relevant, and the study combines the precision of a classical algorithm with the intelligence of modern machine learning approaches. The research is methodologically sound. The authors enhance McSplit by incorporating node embeddings, introduce new heuristics, and compare different selection strategies. A large number of graph pairs are considered in the experimental section. The results demonstrate a significant reduction in computation time and recursion depth while maintaining solution quality. However, in order to improve the quality of the manuscript, I would like to draw the authors' attention to the following comments and recommendations:

**Response**
We are profoundly grateful to the reviewer for their perceptive and accurate analysis, clearly articulating our work's core contribution.

***

**Comment 1**
The literature review is rather narrow and lacks critical analysis of previous methods.

**Response 1**
We sincerely appreciate the reviewer's guidance regarding needing a more comprehensive literature review. We have significantly expanded the introduction section, now meticulously structured into several subsections. Notably, Section 1.1 is now entirely dedicated to related work. This expanded section includes more related works, increasing from 25 in the original version to 34 in the current one. We have included a critical discussion of their main limitations for each of the most pertinent works, aiming for enhanced clarity and a more robust contextualization of our work within the existing literature. The final paragraph of this section now clearly delineates our method's principal differences and contributions compared to these existing works.

***

**Comment 2**
Although the results are presented in detail, the authors are encouraged to summarize the key indicators in a dedicated summary table to improve formal clarity and facilitate understanding.

**Response 2**
We are grateful for this insightful suggestion to enhance the clarity of our results.
First, we have substantially increased the number of graph pairs used in our experiments, expanding from 350 to 500. This expansion allowed us to incorporate more challenging graph pairs, which, in turn, better highlight the distinct advantages of our method compared to previous approaches. Furthermore, we have reorganized the experimental section into several different subsections (from Section 3.3 to 3.7) and merged each pair of tables into a single, cohesive table to improve the logical flow and ease of comprehension. Finally, Section 3.8 now features Figure 7, a comprehensive table summarizing our results, and three insightful pie plots that graphically represent the superior performance of our strategies against those referenced.

***

**Comment 3**
The statement in the conclusions regarding "significant improvement" is not statistically supported. I recommend including at least a basic analysis of confidence or distribution to justify the claim.

**Response 3**
We appreciate this crucial point.

The adjective "significant" in the paper is not used as "statistically significant", but rather means "noteworthy" from a practitioner's point of view: We improved the quality of the results under a limited time budget, and this could be pretty relevant for industrial applications.

Moreover, our algorithms are designed to be practically deterministic, and we have consistently observed the same performance with a remarkably tight Confidence Interval. This high confidence level rigorously underscores the reliability and accuracy of our results. More specifically, before the initial submission, we extensively tested our algorithms on a dataset of 350 graphs. We consistently found no significant performance differences when modifying, enlarging, or randomly altering the dataset. For this revised submission, we further expanded our experiments to include 500 pairs of graphs to validate our findings robustly. We intentionally introduced graph pairs with a higher degree of difficulty and are pleased to report that all previous results have been generally confirmed and, in many cases, even improved.

Furthermore, it is essential to note that each of our experiments runs for a maximum of 3,000 seconds. This time implies that for 500 pairs of graphs, and considering the various algorithms and configurations tested (McSplit, McSplitRL, Norm for K=1-8 across 8 cases, and CCS for K=1-8 across 8 cases), the upper bound for experimental time is substantial and estimated at approximately 7,500 hours of CPU time (i.e., 312 days). Fortunately, many graph pairs are solved well before the 3,000-second limit.
Such extensive experimental costs were primarily managed during the original preparation of the paper, with minor extensions for this second version, leveraging a prolonged period of experimental analysis across several parallel tasks and workstations.

***

**Comment 4**
The reference list requires revision. Several sources do not contain DOIs. In addition, reference [25] (Private Communication) cannot be considered a valid scientific reference.

**Response 4**
We have meticulously revised the reference list to include all previously missing information. Furthermore, we have added several new references, increasing the total count from 25 to 34 entries. Regarding old reference [25] (Private Communication), its inclusion was not intended to signify a valid scientific reference but rather to clarify the challenges we faced in comparing our data with GLSearch (now reference [29]):

Yunsheng Bai, Derek Xu, Yizhou Sun, and Wei Wang, "GLSearch: Maximum Common Subgraph Detection via Learning to Search", Proceedings of the 38th International Conference on Machine Learning, pp. 588-598, 2021 [cite: 43]

Despite numerous contacts and email exchanges with the GLSearch development group members, we unfortunately could not train their model from scratch or run their public implementation against different datasets. We have now inserted the reference and the comments within the paper into a footnote, Footnote 4 on Page 15.

We invite the reviewer to refer to Response 4 to Reviewer 2 for further details.

***

**Comment 5**
Furthermore, there is a substantial risk of self-plagiarism in this work, which should be clearly acknowledged and explained by the authors.

**Response 5**
We acknowledge the reviewer's concern regarding potential self-plagiarism.
While we have indeed been engaged in graph-related challenges for a considerable time, our prior publications specifically on the MCS problem are limited to the following two papers:

1. Stefano Quer, Andrea Marcelli, Giovanni Squillero, "The Maximum Common Subgraph Problem: A Parallel and Multi-Engine Approach", Computation, Publisher MDPI AG, Switzerland, Vol. 8, N. 2, Article Number. 48, 2020, pp. 1-29, https://www.mdpi.com/2079-3197/8/2/48/pdf](https://www.mdpi.com/2079-3197/8/2/48/pdf, ISSN 2079-3197, DOI: 10.3390/computation8020048

This work focuses on parallel and multi-engine (portfolio) computation.

2. Lorenzo Cardone, Stefano Quer, "The Multi-Maximum and Quasi-Maximum Common Subgraph Problem", Computation, Publisher MDPI AG, Switzerland, vol. 11, No. 4, 2023, pp. 69-94, https://www.mdpi.com/2079-3197/11/4/69, https://www.mdpi.com/2079-3197/11/4/69, ISSN: 2079-3197, DOI: 10.3390/computation11040069

This paper addresses the multi-MCS problem (i.e., finding the MCS on more than two graphs).

Certain introductory, background, and related works sections may share similarities with previous papers on the same topic. As the reviewer may agree, the fundamental definition of a graph, perhaps drawn from standard references like "Introduction to Algorithms" by T. H. Cormen et al., will inevitably coincide or be very similar across work. Crucially, none of our previous publications incorporates machine learning techniques into the standard MCS function computation. Therefore, the core algorithmic contributions and the entirety of the experimental section in the current submission are entirely novel and do not overlap with our prior works.
These considerations unequivocally ensure the originality of the current submission.

Nonetheless, we are grateful to the reviewer for highlighting the potential for self-plagiarism, and we have meticulously reviewed the language style in the introductory parts to ensure distinctiveness.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper proposes an innovative method that significantly improves the performance of the McSplit algorithm in MCS problems by combining GNN and node embedding techniques. Experimental results show that this method outperforms existing methods in most cases, especially on large-scale graphs. However, the method details, experimental design, and result analysis still need further refinement.

1. The paper mentions that recalculating node embeddings during the recursive process significantly increases computational cost, but does not delve into whether more efficient update strategies exist. It is recommended to supplement this content or discuss how to balance computational cost and performance improvements, similar to the methods in “Parallel Computing of Spatio-Temporal Model Based on Deep Reinforcement Learning.”

2. The experimental range for k is from 1 to 8, but the rationale for selecting this range is not explained. It is recommended to further analyze the impact of k values on embedding quality and computational efficiency, and provide theoretical or experimental support.

3. The paper directly uses the pre-trained NeuroMatch model but does not discuss whether it is fully applicable to the MCS problem. The authors should clarify whether the dataset and task of the pre-trained model are relevant to the MCS problem or whether fine-tuning is required for MCS.

4. The experiments do not include recent methods such as GLSearch, with the rationale that GLSearch is only applicable to connected subgraph problems. It is recommended to discuss the limitations of GLSearch or other related methods and explain the advantages of the proposed method in terms of generality.

5. CCS outperforms L2 norm in the experiments, but the fundamental reasons are not thoroughly analyzed. It is recommended to explain why CCS is more effective from a theoretical or experimental perspective, such as whether it is related to its ability to capture global structure, referencing “Prediction of Cancellation Probability of Online Car Hailing Order Based on Multi-source Heterogeneous Data Fusion.”

6. Some terminology is inconsistent, such as “CSS” and ‘CCS’ (cumulative cosine similarity). It is recommended to standardize it as “CCS.”

Author Response

We sincerely thank the editor and reviewers for their meticulous review and insightful suggestions. We have diligently addressed all comments in this revised version of the paper. We hope that the same Reviewers will review the paper and that our adjustments will satisfy them.

From a high-level perspective, our key revisions include:

• A comprehensive rectification of all major stylistic and English language issues, achieved through a careful revision and a pass with Grammarly Pro.
• A deeper analysis, from both the theoretical and experimental points of view, of the norm and cosine similarity approaches, their comparison, the NeuroMatch model, and the selection of the primary vital parameters (such as k).
• Additional experiments on challenging graph pairs to demonstrate the advantages of our proposed method more vividly. To underscore this, we have modified and incorporated new pie plots in Section 3.8, Figure 7, on Page 21, with a deeper description of the numerical advantages of our methods.
• A complete overhaul of the reference list, now including a more in-depth analysis of related papers. This action has expanded our reference count from 25 to 34.

The new version of the paper is approximately four pages longer than the original submission. We have highlighted all major modifications in blue within the paper for easy identification, intentionally omitting minor changes such as typos or small corrections.
The following sections provide explicit discussions of issues that required more than superficial corrections, presented as direct answers to the reviewers' comments.

***

**General Comment**
This paper proposes an innovative method that significantly improves the performance of the McSplit algorithm in MCS problems by combining GNN and node embedding techniques. Experimental results show that this method outperforms existing methods in most cases, especially on large-scale graphs. However, the method details, experimental design, and result analysis still need further refinement.

**Summary Response**
We are profoundly grateful for the reviewer's perceptive and accurate analysis, which eloquently articulates the core contribution of our work.

***

**Comment 1**
The paper mentions that recalculating node embeddings during the recursive process significantly increases computational cost, but does not delve into whether more efficient update strategies exist. It is recommended to supplement this content or discuss how to balance computational cost and performance improvements, similar to the methods in "Parallel Computing of Spatio-Temporal Model Based on Deep Reinforcement Learning."

**Response 1**
We appreciate the reviewer's suggestion regarding more efficient embedding update strategies.
The following referenced article:

Zhiqiang Lv, Jianbo Li, Zhihao Xu, Yue Wang, Haoran Li, "Parallel Computing of Spatio-Temporal Model Based on Deep Reinforcement Learning", 2021, isbn: 978-3-030-85927-5, Springer-Verlag, Berlin, Heidelberg, https://doi.org/10.1007/978-3-030-85928-2_31, doi = {10.1007/978-3-030-85928-2_31}

effectively optimizes deep learning models for spatio-temporal data through data and model parallelism, and a gradient accumulation algorithm.
Our approach leverages graph neural networks to enhance state-of-the-art "sequential" MCS branch-and-bound algorithms with dynamic sorting heuristics based on node embedding. We deliberately focused on this core contribution as the MCS problem is intrinsically complex to parallelize and essentially work-unbalanced. Implementing efficient multi-threading (CPU) or many-threading (GPU) of the MCS problem is exceptionally complex. Therefore, we consider it beyond the scope of the current paper.

However, to address the reviewer's valuable feedback and pave the way for future exploration, we have revised the paragraphs describing our task-parallel approach in Section 2.4 (please see the blue text on Pages 13 and 14). The strategy, albeit trivial, delivers excellent results, and we hope it sufficiently motivates the paper's content.

Moreover, we have completely rewritten the conclusion section, significantly enhancing the discussion on "smart" parallelism techniques future work.

***

**Comment 2**
Given an anchor node v, the value of k controls the depth of the breadth-first visit used to build our k-hop neighborhood subgraphs. The experimental range for k is from 1 to 8, but the rationale for selecting this range is not explained. It is recommended to further analyze the impact of k values on embedding quality and computational efficiency, and provide theoretical or experimental support.

**Response 2**
As the reviewer accurately points out, for a given anchor node v, the value of k dictates the depth of the breadth-first search used to construct our k-hop neighborhood subgraphs. Theoretically, a larger k is expected to yield a more precise estimate and, consequently, superior node ordering.
However, as demonstrated in Tables 1, 2, 3, and 4, our empirical findings reveal a diminishing return: Increasing k beyond a certain point offers no additional improvements in the number of graph pairs solved. Given that the computational cost of our heuristic increases proportionally with k, we empirically determined that k=8 serves as an optimal threshold, beyond which further graph analysis provides no significant benefits. We may argue that analyzing nodes too distant from the anchor node may introduce interdependencies among node selections, thereby potentially nullifying the unique contributions of individual node analyses.

We have thoroughly clarified these points in the revised version of the paper, with detailed explanations highlighted in blue in Section 2.4 on Page 13.

***

**Comment 3**
The paper directly uses the pre-trained NeuroMatch model but does not discuss whether it is fully applicable to the MCS problem. The authors should clarify whether the dataset and task of the pre-trained model are relevant to the MCS problem or whether fine-tuning is required for MCS.

** Response 3**
We thank the reviewer for highlighting this critical point regarding the applicability of the pre-trained NeuroMatch model to the MCS problem. We have incorporated a detailed discussion in Section 3.1 to address this concern (please see the blue text on Pages 14 and 15). The key points of this addition are summarized below.

We selected the NeuroMatch model due to its significant conceptual similarities with the MCS problem. NeuroMatch is primarily trained to learn subgraph embeddings that satisfy strong subgraph relationship constraints (such as transitivity, anti-symmetry, and intersection), enabling it to encode
k-hop structural information for nodes in an order-preserving manner. The original NeuroMatch pre-training utilized a diverse combination of synthetic and real-world graph datasets, focusing on the general task of subgraph matching and inclusion, rather than exclusively on MCS instances. Crucially, its loss function is specifically designed to distinguish when one graph is a subgraph of another, a capability highly relevant to the MCS task, where the objective is to discover large common subgraphs.

The pre-trained model demonstrated robust performance on standard MCS benchmarks (unlabeled, undirected graphs) in our experiments. We did not perform further fine-tuning, as our approach leverages these embeddings as feature vectors to rank and select node matches, and does not require supervision specific to MCS instances. An off-the-shelf model is advantageous, as it suggests that more domain-specific training or fine-tuning (e.g., MCS solution data) could yield even further improvements. NeuroMatch's architecture is model-agnostic and compatible with other graph neural network backbones and training regimes, allowing for its adaptation to new graph domains or MCS variants.

Therefore, while the NeuroMatch pre-training is not MCS-specific, its subgraph-centric training regime, order-preserving embeddings, and proven transferability to subgraph discovery tasks strongly support its applicability to the MCS heuristics presented in our work. We acknowledge the significant potential for domain-specific fine-tuning, particularly for labeled, directed, or structurally distinct graphs, and intend to explore this further in our future work.

***

**Comment 4**
The experiments do not include recent methods such as GLSearch, with the rationale that GLSearch is only applicable to connected subgraph problems. It is recommended to discuss the limitations of GLSearch or other related methods and explain the advantages of the proposed method in terms of generality.

**Response 4**
We appreciate the opportunity to clarify our reasons for not including experiments with GLSearch in our paper, a point we briefly touched upon in our initial submission. Our primary rationale stems from significant practical challenges: despite extensive efforts to contact the development group (as noted in our "Private Communication" reference), we could not successfully train their model from scratch.
Furthermore, attempts to run their publicly available implementation on datasets beyond those used by the original authors proved either impossible or consistently yielded significantly flawed or incorrect results.
To provide additional context, we also endeavored to compare our applications against other Reinforcement Learning-based variants. However, these approaches consistently exhibited limitations, primarily leading to the identification of only suboptimal or small solutions.
At the same time, all "traditional" techniques (i.e., without ML-based improvements) derived from McSplit present similar results. Therefore, we restrict our comparison to the original McSplit as representative of the traditional technique and McSplitRL as representative of the conventional technique boosted by ML-based approaches.

For further details concerning the topic and the "Private Communication" reference (old reference [25]), please refer to Response 4 to Reviewer 1.

***

**Comment 5**
CCS outperforms L2 norm in the experiments, but the fundamental reasons are not thoroughly analyzed. It is recommended to explain why CCS is more effective from a theoretical or experimental perspective, such as whether it is related to its ability to capture global structure, referencing "Prediction of Cancellation Probability of Online Car Hailing Order Based on Multi-source Heterogeneous Data Fusion."

**Response 5**
We sincerely thank the reviewer for directing our attention to the article

Haokai Sun, Zhiqiang Lv, Jianbo Li, Zhihao Xum, Zhaoyu Sheng, Zhaobin Ma, "Prediction of Cancellation Probability of Online Car-Hailing Orders Based on Multi-source Heterogeneous Data Fusion, 2022, pp. 168-180, Springer-Verlag, Berlin, Heidelberg, https://doi.org/10.1007/978-3-031-19214-2_14, ISBN: 978-3-031-19213-5, DOI: 10.1007/978-3-031-19214-2_14

This work, which predicts online car-hailing order cancellation probability, compellingly demonstrates that fusing diverse and context-rich information leads to more accurate predictions. This principle directly extends to the MCS problem: the superiority of Cumulative Cosine Similarity (CCS) over the L2 norm likely stems from its inherent ability to "fuse information across both graphs", acting as a "multi-source" approach to generate a more comprehensive and "globally aware" heuristic score for node pairing.
Rather than relying solely on a node's isolated structural characteristics, CCS captures more nuanced "compatibility" between nodes. This capability leads to more effective search space pruning and, consequently, faster identification of larger common subgraphs.

We have incorporated this crucial consideration into Section 3.5 on Page 18 of the revised manuscript.

***

**Comment 6**
Some terminology is inconsistent, such as "CSS" and 'CCS' (cumulative cosine similarity). It is recommended to standardize it as "CCS."

**Response 6**
You are correct regarding this inconsistency.
We apologize for the oversight and have thoroughly standardized the terminology throughout the manuscript, consistently using the abbreviation "CCS" for Cumulative Cosine Similarity

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have carefully addressed all major concerns raised in the previous review. The revised version presents a significantly expanded literature review, a more precise articulation of the methodological novelty, and a more structured and detailed experimental analysis.

The current manuscript demonstrates both theoretical originality and practical relevance.