Paper: FL-GNN: A Fuzzy-logic Graph Neural Network (ICLR-2024)

Reviewer comment (Experiments):
While the paper claims that FL-GNN outperforms other GNN models, it could benefit from a more rigorous benchmarking, including challenges faced, methodologies adopted, and potential pitfalls.

Author response:
**RE:** Thank you for your valuable suggestions. In the revised manuscript, we summarize the advantages of FL-GNN compared to traditional GNNs in Section 1 on page 2.

**(1) Representation capacity.**
Fuzzification has been extensively exploited to improve data representation capabilities. According to the Introduction Section and two supporting references [1,2], real-world data often contains uncertainty and fuzziness beyond the scope of traditional crisp values. To tackle this challenge, tools for fuzzification have been developed to capture such fuzziness and uncertainty. However, relying solely on fuzzification as a data augmentation technique is insufficient to fully utilize the benefits of fuzziness. Traditional GNN frameworks are limited in their feature extraction, which focuses solely on feature dimensions and disregards the fuzzy dimension. To overcome this limitation, we have created the FL-GNN, which utilizes fuzzy rules to highlight fuzzy-level features and leverages the advantages of fuzzy features for a more comprehensive data representation solution.

**(2) Interpretability.**
In traditional GNNs, the correlation between the network's parameter weights and inference process is implicit. In FL-GNN, the differences in topological structure between communities are explicitly presented in the firing strength distribution. By studying the firing strength distribution, we can know which rules are valid and which are redundant. Therefore, in this paper, interpretability not only provides a visual perspective for us to observe the local topology information and feature distribution differences of vertices but also provides reliable evidence for us to improve the model structure.

In Appendix C, we provide additional experiments and analyses including: 1) Interpretability of indirect interactions between nodes. 2) Interpretability experiments conducted on the real-world dataset Pubmed. 3) The interpretability comparison between FL-GNN-A and GNNExplainer[3].

**(3) Degree of freedom.**
Several frameworks have been developed in graph inference, such as Message Passing Algorithms, which are compatible with most graph inference models. Fuzzy inference systems can offer a more flexible way to establish inference models. Fuzzy rules are not limited to graph-structured data and can be extended to other data structures and temporal data processing [4]. In addition, in [2], fuzzy rules are directly designed for specialized graph-level task inference. Therefore, we believe that the potential of FL-GNN will be fully realized in future works, where more effective fuzzy rules can be designed for graph inference tasks. In Appendix A, we have mentioned that the fuzzy rule of FL-GNN is limited to message aggregation in 1-order neighbors. Nevertheless, by designing more sophisticated fuzzy rules, they can be extended to complex topology structures such as "Hyperedge" and "Simplicial Complex", which will help the model break through the 1-dim WL-test.

[1] [Fuzzy Representation Learning on Graph, IEEE Trans. Fuzzy Syst. 2023.](https://ieeexplore.ieee.org/document/10061283)

[2] [Graph Fuzzy System for the Whole Graph Prediction: Concepts, Models and Algorithms, IEEE Trans. Fuzzy Syst. 2023.](https://ieeexplore.ieee.org/document/10287583)

[3] [GNNExplainer: Generating Explanations for Graph Neural Networks, NeurIPS 2019.](https://arxiv.org/abs/1903.03894)

[4] [An evolving recurrent interval type-2 intuitionistic fuzzy
neural network for online learning and time series prediction, Appl. Soft Comput 2019.](https://doi.org/10.1016/j.asoc.2019.02.032)

Reviewer comment (Presentation):
It would benefit the paper to clearly outline the limitations of the proposed FL-GNN, and how these have been addressed, to provide a comprehensive perspective.

Author response:
**RE:** Thank you for your constructive suggestions. Regarding Question 2, we have added the differences between Type-2 fuzzy sets and intuitionistic fuzzy sets, as well as Type-1 fuzzy sets used in FL-GNN in Section 5 on page 10. Regarding weakness 2, we describe the aspects that may limit FL-GNN to reach its full potential in Section 5:

1\) FL-GNN uses Type-1 fuzzy sets to fuzzify the feature data of vertices, but the ability of Type-1 fuzzy sets to represent fuzziness is limited. Instead, more powerful fuzzy sets such as Type-2 fuzzy sets and intuitionistic fuzzy sets can provide stronger fuzziness. By using these fuzzy sets, we can obtain better representation capabilities.

2\) Computational complexity is relatively high given that FL-GNN needs to handle a large number of fuzzy rules. Though we proposed FL-GNN-A, which can greatly alleviate this problem and enable FL-GNN-A to work effectively on large datasets. One of our ongoing researches is to develop optimizations towards further improvement of the computational complexity. One direction we plan to pursue is to use the interpretability property of FL-GNN to guide us in designing evolutionary algorithms to compress network structures and obtain more effective models.

Reviewer comment (Evaluation):
In addition to benchmarking performance, a detailed comparative analysis on how FL-GNN overcomes the limitations of standard GNNs would be insightful.

Author response:
**RE:** Thank you for your valuable suggestions. In the revised manuscript, we summarize the advantages of FL-GNN compared to traditional GNNs in Section 1 on page 2.

**(1) Representation capacity.**
Fuzzification has been extensively exploited to improve data representation capabilities. According to the Introduction Section and two supporting references [1,2], real-world data often contains uncertainty and fuzziness beyond the scope of traditional crisp values. To tackle this challenge, tools for fuzzification have been developed to capture such fuzziness and uncertainty. However, relying solely on fuzzification as a data augmentation technique is insufficient to fully utilize the benefits of fuzziness. Traditional GNN frameworks are limited in their feature extraction, which focuses solely on feature dimensions and disregards the fuzzy dimension. To overcome this limitation, we have created the FL-GNN, which utilizes fuzzy rules to highlight fuzzy-level features and leverages the advantages of fuzzy features for a more comprehensive data representation solution.

**(2) Interpretability.**
In traditional GNNs, the correlation between the network's parameter weights and inference process is implicit. In FL-GNN, the differences in topological structure between communities are explicitly presented in the firing strength distribution. By studying the firing strength distribution, we can know which rules are valid and which are redundant. Therefore, in this paper, interpretability not only provides a visual perspective for us to observe the local topology information and feature distribution differences of vertices but also provides reliable evidence for us to improve the model structure.

In Appendix C, we provide additional experiments and analyses including: 1) Interpretability of indirect interactions between nodes. 2) Interpretability experiments conducted on the real-world dataset Pubmed. 3) The interpretability comparison between FL-GNN-A and GNNExplainer[3].

**(3) Degree of freedom.**
Several frameworks have been developed in graph inference, such as Message Passing Algorithms, which are compatible with most graph inference models. Fuzzy inference systems can offer a more flexible way to establish inference models. Fuzzy rules are not limited to graph-structured data and can be extended to other data structures and temporal data processing [4]. In addition, in [2], fuzzy rules are directly designed for specialized graph-level task inference. Therefore, we believe that the potential of FL-GNN will be fully realized in future works, where more effective fuzzy rules can be designed for graph inference tasks. In Appendix A, we have mentioned that the fuzzy rule of FL-GNN is limited to message aggregation in 1-order neighbors. Nevertheless, by designing more sophisticated fuzzy rules, they can be extended to complex topology structures such as "Hyperedge" and "Simplicial Complex", which will help the model break through the 1-dim WL-test.

[1] [Fuzzy Representation Learning on Graph, IEEE Trans. Fuzzy Syst. 2023.](https://ieeexplore.ieee.org/document/10061283)

[2] [Graph Fuzzy System for the Whole Graph Prediction: Concepts, Models and Algorithms, IEEE Trans. Fuzzy Syst. 2023.](https://ieeexplore.ieee.org/document/10287583)

[3] [GNNExplainer: Generating Explanations for Graph Neural Networks, NeurIPS 2019.](https://arxiv.org/abs/1903.03894)

[4] [An evolving recurrent interval type-2 intuitionistic fuzzy
neural network for online learning and time series prediction, Appl. Soft Comput 2019.](https://doi.org/10.1016/j.asoc.2019.02.032)

Reviewer comment (Writing):
The introduction of terms such as Type-2 fuzzy sets and intuitionistic fuzzy sets, which are mentioned as future work, might be unfamiliar to some readers. A brief description or reference would be helpful.

Author response:
**RE:** FL-GNN is compatible with Type-2 fuzzy sets. Here, we use a simple example to illustrate. Usually in FNN, in order to reduce the computational complexity, we use interval Type-2 fuzzy sets [1]. In interval Type-2 fuzzy sets, we can use two Type-1 fuzzy sets, i.e., lower membership function (LMF) and upper membership function (UMF), to characterize the upper and lower bounds of the membership function, respectively. The second-order fuzziness of interval Type-2 fuzzy sets is reflected by the gap between UMF and LMF, i.e., the footprint of uncertainty (FOU). If the FOU disappears, the interval Type-2 fuzzy sets will degenerate into Type-1 fuzzy sets. For the FL-GNN that introduces interval Type-2 fuzzy sets, we only need to calculate the firing strength of each fuzzy rule on LMF and UMF to obtain $\overline{\pmb{r}_n}=[\overline{r^1_n}, \overline{r^2_n}, \dots, \overline{r^K_n}]$ and $\underline{\pmb{r}_n}=[\underline{r^1_n}, \underline{r^2_n}, \dots, \underline{r^K_n}]$ at the rule layer, where $n$ is the index of the node in the graph and $K$ is the number of fuzzy rules. $\overline{\pmb{r}_n}$ is the firing strength vector for UMF and  $\overline{r^k_n}$ is the firing strength of the $k$th fuzzy rule of UMF, $\underline{\pmb{r}_n}$ is the firing strength vector for LMF and $\underline{r^k_n}$ is the firing strength of the $k$th fuzzy rule of LMF. 
Next, we can simply calculate the median value $\pmb{r}_n= \frac{\overline{\pmb{r}_n} + \underline{\pmb{r}_n}}{2}$ as the final firing strength of the $k$th fuzzy rule. Alternatively, we can directly concatenate $\pmb{r}_n= [\overline{\pmb{r}_n}, \underline{\pmb{r}_n}]$ and input them into the MLP to transform them into the form required by the downstream task, because $\overline{\pmb{r}_n}$ and $\underline{\pmb{r}_n}$ both possess the node feature information and graph topology information needed for graph inference.

With the introduction of the Type-2 fuzzy set, the complexity of the model will be further increased compared to the Type-1 fuzzy set. The key challenge is how to reduce the computational complexity of the model. Meanwhile, to better capture the second-order fuzziness of the data, how to set the initial FOU of UMF and LMF and maintain the effective FOU during model training are also the key challenges we will face.

[1] [Type-2 Fuzzy Logic: Theory and Applications, 2008](https://dblp.org/rec/books/sp/CastilloM08.html?view=bibtex)

Reviewer comment (Experiments):
Some claims need to be further validated. The authors argue that the interpretability of FL-GNN is good while they ignore the experiments to compare with some graph interpretability baselines (please refer to [1]).

Author response:
**RE:**  Thanks for your comments and suggestions. FL-GNN's interpretability is inherent in fuzzy inference systems. The paper shows formally and empirically that FL-GNN can retain this interpretability in graph inference in specific scenarios.

Following your suggestion, we provide additional experiments and analysis in Appendix C to validate our claims on interpretability, including: 1) Interpretability of indirect interactions between nodes; 2) Interpretability experiments conducted on the real-world dataset Pubmed; 3) Interpretability comparison between FL-GNN-A and GNNExplainer [1].

[1] [GNNExplainer: Generating Explanations for Graph Neural Networks, NeurIPS 2019.](https://arxiv.org/abs/1903.03894)

Reviewer comment (Experiments):
The experiments in terms of node-level predictions should consider more large-scale datasets from OGB, just as the authors have done in graph-level predictions.

Author response:
**RE:** Thanks for your suggestion. We have provided an experiment on a large-scale node-level dataset "ogbn-proteins" in Section 4.2 of the revised manuscript. The new experimental results are displayed in Table 2-1 (a part of Table 1 in the revised manuscript), showing that our model achieved comparable results to traditional GNNs.

&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;**Table 2-1: The experimental results on ogbn-proteins**

|FL-GNN-A|Graphsage-mean|GAT|GCN|TransformerConv|GATv2|GIN|
|------------|------------|------------|------------|------------|------------|------------|
|0.7989&pm;0.0043| 0.7962&pm;0.0039|0.7996&pm;0.0018|0.7551&pm;0.0079 |0.8003&pm;0.0025|0.7908&pm;0.0033|0.7569&pm;0.0054|

Reviewer comment (Writing):
Typos: dataset name Coar -> Cora in section 4.1.

Author response:
**RE:** 
Thank you for your careful review and attention to detail. We sincerely apologize for any inconvenience caused by these typos. We thoroughly revised the article to address these typos and ensure its accuracy and clarity. We proofread the manuscript to correct these typos, e.g.,

- On page 1: "fields such as medical image process" $\rightarrow$  FNN has been successfully applied in various fields such as medical image processing.
- On page 2: "information are ubiquitous" $\rightarrow$ This sentence was deleted after reorganizing this part.
- In experiments: "Coar" $\rightarrow$ "Cora".

Reviewer comment (Experiments):
The paper does not provide convincing evidence that the proposed FL-GNN architecture has consistent, measurable advantages over existing GNN architectures. The experimental results show that FL-GNNs are on-par with previously proposed approaches. While the authors allude to potential insights into graph inference and improved model interpretability in the conclusion, the anecdotal evidence provided in appendix C is, by itself, not sufficient to claim that FL-GNNs provide interpretable predictions. A more comprehensive evaluation and a comparison with general gradient-based approaches, such as [GradCAM](https://arxiv.org/abs/1610.02391), or graph-specific approaches, such as [GnnExplainer](https://arxiv.org/abs/1903.03894) would have been interesting. As it stands, the relevance of the proposed architecture is unclear.

Author response:
**RE:** One of the main contributions made by FL-GNN is its interpretability inherent in fuzzy inference systems. This paper mainly intends to demonstrate that FL-GNN can retain this interpretability in graph inference in specific scenarios from the following three aspects: the meaning and the potential value of FL-GNN interpretability, and the comparative discussion with the existing representative methods, such as [1-2] provided by the reviewer.

**(1) The meaning of FL-GNN's interpretability.**

First, as discussed in Appendix C, in a social network scenario where the raw features of each vertex are comprehensible, with FL-GNN, one can confer the feature with meaningful linguistic labels to make the fuzzy rule intuitively understood. 

Second, the interpretability of FL-GNN is not limited to linguistic labels and can capture certain meanings based on hidden features. That is, the firing strength distribution of the fuzzy rule can provide interpretations for the inference process of FL-GNN. The firing strength distribution of communities directly reflects their topological structure information and feature distribution. Here, topological information is specified as the degree of community overlap, i.e., the probability of having edges within a community and between multiple communities. Feature distribution specifically refers to the differences in node feature value between different communities. As shown in Appendix C, the communities of different professions exhibit intuitive differences in firing strength distribution. 

**In FL-GNN, the differences in topology structure between communities are explicitly presented in the firing strength distribution, indicating that there is a direct relationship between which fuzzy rules have a positive effect on capturing topology differences and which ones do not work.** Therefore, our improvement in fuzzy rules can be intuitively reflected in the differences in model performance, just as we performed MaxPooling on the output of fuzzy rule layers in FL-GNN-A. Filtering out redundant rules not only ensures model accuracy but also improves the inference speed of the model.

**(2) The potential value of FL-GNN's interpretability.**

We believe that studying firing strength distributions has significant potential values because the fuzzy rule provides a medium that allows us to indirectly explain the working principles of neural networks by directly observing their intermediate results. The firing strength distributions can make uncovering flaws in the network structure or the dataset easier. Here, we provide several possible explorations that can take advantage of the FL-GNN's interpretability.

1\) Outlier detection: For example, if there exist some nodes that have a significant difference in their firing strength distributions compared to their communities, or two different communities present resemble firing strength distributions, then there might be misclassification or overclassification of labels in this dataset, and the dataset may also have higher-order topological differences that the current fuzzy rule cannot capture.

2\) Optimizing network structure: In this paper, we have also made a simple attempt to use firing strength distribution to optimize the structure of FL-GNN. In part-2 of FL-GNN-A, we add the MaxPooling layer after observing redundant low firing strength fuzzy rules. In addition, we are considering combining interpretability with evolutionary algorithms to adjust network architecture and make it more efficient.

Reviewer comment (Theory):
The definition of the FL-GNN architecture in section 3 is, in parts, lacking formal accuracy. For example:  
- Section 3.1: The signatures of $\phi$ and $\psi$, as well as those of $\rho$ and $\sigma$ are unclear. The authors state that the signature of $\phi$ is $V \to F_v$ but also that $\phi(v_i) \subset F_v$. Similarly, supposedly $\rho(\phi(v_i)) \subset A_v$, even though the target domain of $\rho$ is $A_v$, not $2^{A_v}$.  
- Section 3.2, eq. 10: Why is the "AND" in the condition taken over all $A_m \in \{ A_{m_{1,i}},\dots, A_{m_{D,j}} \}$? Since $i$ is used to denote the feature index and $j$ to denote the index of a fuzzy subset, $A_{m_{1,j}}$ was probably meant here. Is this correct?  
- Section 3.2: $S$ is defined as the Cartesian product of $D$ sets of size $M$, therefore $|S| = M^D$; in the last line on page 4 the authors do however state that $S$ has $M^D$ subsets.  
Even though each of these examples can be considered to be minor formal mistakes, the repeated unclear usage of "element" and "subset" of a set make it difficult to understand the precise meaning of the provided formal definitions.

Author response:
**RE:** Thanks for your helpful comments. We have corrected the errors in the revised manuscript.

Subquestion 1
-

**RE:** The mapping functions were defined inaccurately, and we corrected them as follows:

Let $\phi: V \rightarrow 2^{F_v}$, $\psi: E \rightarrow 2^{F_e}$ denote the mapping functions for vertex attributes and edge attributes, where $2^{F_v}$ and $2^{F_e}$ are the power sets of $F_v$ and $F_e$, respectively, 
and then each vertex $v_i \in V$ and edge $e_i \in E$ can be represented by a set of vertex attributes $\phi(v_i) \subset F_v$ and a set of edge attributes $\psi(e_i)\subset F_e$. Meanwhile, we also define two fuzzy subset mapping functions, $\rho: 2^{F_v} \rightarrow 2^{A_v}, \sigma: 2^{F_e} \rightarrow 2^{A_e}$.

Subquestion 2
-

**RE:** For the first question, the calculation result of two fuzzy subsets by using logic conjunction operation ("AND") or logic disjunction operation ("OR") is still a fuzzy subset, and the expression $v\ is\ A_m$ is a procession of fuzzification that also generates a fuzzy subset. 
the expression $v \ is \ A_m$ represents a fuzzy subset generated by fuzzification. In the IF-part of Eq. 10, all operations are performed between fuzzy subsets, so it is reasonable to use the "AND" operation to combine a set of fuzzy subsets. This mathematical expression is commonly used in papers on fuzzy logic [1,2,3] (some papers may use $\bigwedge$ or $T(\cdot)$ to denote the "AND" operation). 

For the second one,  we apologize for misusing indexes $i$ and $j$ in Eq. 10. The indexes $i$ and $j$ in Eq. 10 are unrelated to the feature index and the fuzzy subset index, and they denote one index combined result generated from the Cartesian product. To avoid confusion, we substitute $i$ and $j$ in Eq. 10 with other irrelevant symbols (e.g., $a$ and $b$) in the revised manuscript. 

[1] [Disjunctive Fuzzy Neural Networks: A New Splitting-Based Approach to Designing a T–S Fuzzy Model, IEEE Trans. Fuzzy Syst. 2020.](https://ieeexplore.ieee.org/document/9264752/)

[2] [Hybrid Learning for Interval Type-2 Intuitionistic Fuzzy Logic Systems as Applied to Identification and Prediction Problems, IEEE Trans. Fuzzy Syst. 2018.](https://ieeexplore.ieee.org/document/8286852)

[3] [Deep Takagi–Sugeno–Kang Fuzzy Classifier With Shared Linguistic Fuzzy Rules, IEEE Trans. Fuzzy Syst. 2017.](https://ieeexplore.ieee.org/document/7984865)


Subquestion 3
-
**RE:** We apologize for the minor formal mistakes, where the result of n-ary Cartesian product should be denoted as a set of n-tuples, not subsets. Here, we present an example to clarify this problem. Assuming that we have three sets of fuzzy subsets and each set has three fuzzy subsets $A_{v,1}=\\{A_{m_{1,1}},A_{m_{1,2}},A_{m_{1,3}} \\},A_{v,2}=\\{A_{m_{2,1}}, A_{m_{2,2}}, A_{m_{2,3}}\\}, A_{v,3}=\\{A_{m_{3,1}},A_{m_{3,2}}, A_{m_{3,3}}\\}$, to use the 3-ary Cartesian product to combine these three sets, we obtain: $S=A_{v,1}\times A_{v,2}\times A_{v,3}=$$\\{$$(A_{m_{1,1}},A_{m_{2,1}},$$A_{m_{3,1}})$$,(A_{m_{1,2}},A_{m_{2,1}},A_{m_{3,1}})$,...,$(A_{m_{1,3}}$$,A_{m_{2,3}}$,$A_{m_{3,3}}$)}, $|S|=8$. Each element in $S$ is a tuple such as $(A_{m_{1,1} },A_{m_{2,1}},A_{m_{3,1}})$. We have rewritten inappropriate expressions in the revised manuscript.

Reviewer comment (Theory):
The proposed approach only considers the direct neighbors of a vertex in each rule. Indirect interactions between vertices therefore are not captured by FL-GNNs, as also alluded to by the authors at the end of appendix C. The authors do not discuss if and how meaningful interpretations could be provided in problem domains where such indirect interactions are important.

Author response:
**RE:** Thank you for bringing up the issue of node indirect interaction. We would like to address your concern and clarify that FL-GNN is designed to effectively capture the indirect interaction of nodes (see Appendix A for details). To achieve higher-order reasoning ability, FL-GNN utilizes a stacked structure. Additional experiments have been included in Appendix C to better showcase how FL-GNN can capture indirect interactions between nodes. We have also demonstrated the meaningful interpretations of the firing strength distribution generated by nodes under high-order interactions. Specifically, we have visualized the firing strength distribution of each layer in FL-GNN with the stacked structure. By observing the changes in trends of firing strength distribution in different communities, we have found that as the interaction order increases, the firing strength distribution of a community gradually changes from reflecting the topology information of a single community to that of multiple overlapping communities. Ultimately, the firing strength distributions of all communities will tend to be consistent, representing the topology information of the entire graph.