Do We Really Need Message Passing in Brain Network Modeling?
Authors: Liang Yang
Yuwei Liu
Jiaming Zhuo
Di Jin
Chuan Wang
Zhen Wang
Xiaochun Cao
Venue: ICML 2025 spotlightposter (2025)
Keywords: graph neural networks
message passing
brain network
quadratic network
Primary area: deep_learning->graph_neural_networks

Abstract: Brain network analysis plays a critical role in brain disease prediction and diagnosis. Graph mining tools have made remarkable progress. Graph neural networks (GNNs) and Transformers, which rely on the message-passing scheme, recently dominated this field due to their powerful expressive ability on graph data. Unfortunately, by considering brain network construction using pairwise Pearson’s coefficients between any pairs of ROIs, model analysis and experimental verification reveal that *the message-passing under both GNNs and Transformers can NOT be fully explored and exploited*. Surprisingly, this paper observes the significant performance and efficiency enhancements of the Hadamard product compared to the matrix product, which is the matrix form of message passing, in processing the brain network. Inspired by this finding, a novel Brain Quadratic Network (BQN) is proposed by incorporating quadratic networks, which possess better universal approximation properties. Moreover, theoretical analysis demonstrates that BQN implicitly performs community detection along with representation learning. Extensive evaluations verify the superiority of the proposed BQN compared to the message-passing-based brain network modeling. Source code is available at [https://github.com/LYWJUN/BQN-demo](https://github.com/LYWJUN/BQN-demo).

Do We Really Need Message Passing in Brain Network Modeling?
Liang Yang' Yuwei Liu'
Jiaming Zhuo
Jin Chuan Wang?
Zhen Wang* Xiaochun Cao
Abstract
1. Introductions
Brain network analysis provides a deep understanding of human brain organizations and assists in neurological dis- ease diagnosis (Fornito et al., 2016; Cui al., 2022). As ceneral-purpose language model complex relationships, the research on graphs has a long history, ranging from graph theory (Bondy & Murty, 2008), to network science (Barabási, 2013), to network embedding (Cui et al., 2019). Along with the concept of the brain network, graph mining is widely used brain network analysis, including concepts, characteristics, and models (Fornito et al., 2016; Chung, 2019). Network motif (Sporns & Kötter, 2004), small-world property (Bassett & Bullmore, 2006), and modularity (Meu- nier et al., 2009) are identified from the perspective of char- acteristics, while the persistent homology (Bendich et al., 2016), community detection (Garcia et al., 2018 18) and null model (Váša & MiKić, 2022) are employed to analyze brain network from the perspective of the model.
Brain network analysis plays critical role in brain disease prediction and diagnosis. Graph mining tools have made remarkable progress. Graph neural networks (GNNs) and Transform- ers, which rely on the message- passing scheme, recently dominated this field due to their powerful expressive ability graph data. Unfortunately, by considering brain network construction using pairwise Pearson's 's coefficients between any pairs of ROIs, model analysis and experimental verifi- cation reveal that the message-pussing under both GNNs and Transformers can NOT be fully ex- plored and exploited. Surprisingly, this paper ob- serves the significant performance and efficiency enhancements of the Hadamard product compared to the matrix product, which is the matrix form of message passing, in processing the brain net- work. Inspired by this finding, a novel Brain Quadratic Network (BQN) is proposed by incor- porating quadratic networks, which possess better universal approximation properties. Moreover, theoretical analysis demonstrates that BQN im- plicitly performs community detection along with representation learning. Extensive evaluations verify the superiority of the proposed BQN com- pared to the peessage-passing passing-based based brain network modeling. Source code is available at https: www.tttb...om..ttttttt BQN- demo.
meessag-passing
Graph Learning (Xia et al., 2021) plays an important role in diverse machine learning fields. (1) For tasks on graph data, Graph Neural Networks (GNNs), which combine graph topology and node attribute for representation, recently dominated this field (Wu et al., 2021b). They follow the message- saasse-passingggg scheme (Gilmer et al., 2017) by propagat- ing node representation the domains without explicit graph structure, such as natu- ral language processing (NLP) and computer vision (CV), implicit graph learning also shows amazing performance. The key component Multi-Heads Self-Attention (MHSA) in Transformer (Vaswani et al., 2017) and its variant ViT (Dosovitskiy et al., 2) constructs a fully connected graph between all token pairs and performs a holistic aggregation (Section 3.3). In conclusion, the pessaag-passing scheme dominates many machine learning fields.
message over the graph. (2) For
'Hebei Province Key Laboratory Big Data Calculation, School of Artificial Intelligence, Hebei University of Technology, Tianjin, China 'Tianjin Key Laboratory of Cognitive Computing and Application, College of Intelligence and Computing, Tianjin University, Tianjin, China "School of Computer Science and Tech- nology, Beijing JiaoToon University, Beijing, China "School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), School of Cybersecurity, Northwestern Polytechnical University, Xi' an, China School of Cyber Science and Technology, Shenzhen Cam- pus Sun Yat-sen sen University, Shenzhen, China. Correspondence to: Jiaming Zhuo <jiaming.zhuo zhuo @ outlook.com>>.
Interactions between brain regions are regarded as the key factors for neural development and disorder analysis. Func- tional MRI (IMRI) provides valuable information for explor- ing connectivity by capturing correlations between signal sequences of brain regions. Thus, it is direct to analyze brain networks, which are often constructed using pairwise Pearson correlation coefficients between any pairs of ROIs, using powerful graph learning tools, such as GNNs (Li et al., 2021; Cui et al., 2022; Bessadok et al., 2023) and
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Do We Really Need Message Passing in Brain Network Modeling?
Transformers (Kan et al., 2022b; Yu et al., 2024). Unfortu- nately, following model analysis and experiments motivate questioning this direct treatment in the brain network.
of functional connectivity within the brain. For example, BrainGNN (Li et al., 2021) utilizes ROI-aware GNNs and specialized pooling operators to identify critical nodes, im- proving the model's interpretability. BrainGB (Cui et al., 2022) systematically assesses various GNN designs on brain network data and introduces a practical pipeline that en- hances GNN applications in brain research. FBNetGen (Kan et al., 2022a) constructs learned task-oriented brain graph for downstream tasks. A-GCL (Zhang et al., 2023) proposes an adversarial self-supervised brain graph neu- ral network by integrating graph contrastive learning with ROIs multiband information. However, these approaches often concentrate on local neighborhood aggregation, po- tentially neglecting the significance of interactions between non-adjacent brain regions.
GNN-based methods constructed node attributes from the brain network topology. which goes against GNNs' need for different types of information, and thus can't be fully exploited GNNs' characteristics (Fig. 1).
Transformers employ the constructed brain network containing& a holistic relationship between all ROIs, as initial token embedding. Thus the necessity to explore holistic relations with Transformers is weak (Fig. 1).
Simple classifier with brain functional connectivity matrix as feature outperforms basic GNNs and Trans- formers (Fig. 2).
To enhance the performance of brain network analysis, the fundamental brain network operator should be updated. The recent breakthrough points out that the quadratic function can implement XOR logic operation and possesses better universal approximation properties compared to the linear function (Fan et al., 2018; 2020). Inspired by this, the quadratic network employed in brain network analysis to obtain the Brain Quadratic Network (BQN). BQN iter- atively performs Hadamard product/clement-vin product between the representation in the previous layer and the initial representation to get a new representation. Theoreti- cal analysis reveals that BQN is cquivalent to the updating rule of a community detection objective function based on nonnegative matrix factorization (NMF) of the adjacency matrix. Since the NMF of the adjacency matrix is widely- used community detection paradigm, the proposed BQN seeks representation, which can capture mesoscopic com- munity structure to reflect brain functional modules. The main contributions of this paper are summarized as follows.
Graph Transformers. Graph Transformers utilized in the analysis of brain networks, with the objective of capturing the holistic interaction of brain regions. Brain- NETTF (Kan et al., 2022b) employs classical Transformer encoders to generate ROI embeddings based on Pearson correlation matrices, along with an orthogonal clustered readout, ALTER (Yu et al., 2024) has specifically designed brain graph Transformer capture long-range dependen- cies among brain regions. ContrastPool (Xu et al., 2024) emphasizes attention on ROIs and subjects to learn con- trast graph, guiding the generation of brain representations through graph pooling. BioBGT (Peng et al., 2025) cap- tures the small-world architecture in brain graphs through network entanglement- based technique, emphasizing the biological characteristies of brain structure.
However, many of these methods often conflate the concepts of brain functional connectivity matrices and region of inter- est (ROI) features. The correlation coefficient matrix serves a dual purpose: it acts as both the adjacency matrix of the brain graph and the feature matrix. This duality contradicts established principles in graph research, where topology and features typically regarded as two distinct types of information. This observation prompts us to design simple yet effective network architecture specifically tailored for brain functional connectivity.
We investigate the rationality of the widely adopted message-passing the brain network analysis.
We propose a simple and effective Brain Quadratic Net- work (BQN) as the fundamental operator, with superior computational efficiency.
We provide rigorous theoretical analysis connecting the proposed BQN with community detection.
3. Preliminaries
The proposed BQN achieves new SOTA on widely- used brain datasets.
This section provides notations, problem definition, and the concepts of GNNs and Transformers.
2. Related Work
3.1. Notations and Problem Definition
Recent advancements in brain network analysis have led to a surge in the application of graph-based learning methods, especially Graph Neural Networks and Graph Transformers.
Brain network, which models the connectivity between ROIs, be represented as a graph G (V, E, X), where V {wi, 02,..., uN) stands for the collection of N nodes, i.c.. ROIS, E denotes the collection of edges between nodes, and X E NDD represents the feature matrix with the
Graph Neural Networks. In recent studies, Graph Neural Networks (GNNs) have been leveraged to learn represen- tations of brain regions, as well as the intricate patterns
2

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Do We Really Need Message Passing in Brain Network Modeling?
Node Attributes
Topology information is used twice
from Topology
Brain Network Construction
Graph Neural
one-hot
Networks
AX
AI
ROMM
Q-XW
K=XW
V-XW
Transformers
ROI2
RONP
Brain Network Topology
Rovv
NOMM
attention mechanisms
Holistic relationship is present in the topology
Figure 1. Brain network construction the GNNs and Transformers on brain network analysis. brain network construction, node attributes are from constructed brain network topology. Thus, GNNs utilize this information twice. The brain network already contains the correlation between all ROIs, while attention mechanism also tends to learn this.
i-th row X; R D as the feature of node/ROI v,. The graph topology is represented as the adjacency matrix A [ayy] NNN, where di is the weight between nodes o and Ug. (u) denotes the neighbourhoods of node v. D E R N stands for the degree matrix with diagonal element d,s- = EveN(v) Que as the degree of node v.
where a denotes the non-linear activation function and H X denotes the initial node attributes. Á = (D +
I, - (A + (D. + I,,) represents the normalized ad- jacency matrix with self-loop.
3.3. Transformers
For brain network analysis tasks, a set of subjects' brain network 9 = [Gi G.} and the corresponding labels )'= (y ... ... y which indicates the presence of a disease, biological sex other properties of the subject, are provided. Brain network modeling aims to learn from given data G and y by designing function = f(h(G)), which is composed of a representation learning function lig (G) and a prediction function y = (ha) based on the learned brain representation lg.
Unlike GNNs that aggregate information from local neigh- borhoods, Transformers facilitate a holistic aggregation across all token pairs through the self-attention mechanism. The main component of Transformer (Vaswani al., 2017) is Multi-Heads Self-Attention (MHSA), Given N tokens input Z = [2,
/=) N-1 E IR NxF, which is the concatenation of the initial token embedding X and the position encoding P as = |X| the self-attention first maps the input features to query (Q), key (K), and value (V) vectors. Then, attention scores from query-key pairs employed aggregate the value vectors in weighted manner, operating as global message-passing mechanism. Specifically, Delf-Attention (SA) module can be formulated as
3.2. Graph Neural Networks
Graph Neural Networks (GNNs) often follow the message- passing paradigm (Gilmer et al., 2017; Yang et al., 2022; Zhuo et al., 2023), by iteratively aggregating node represen- tations from neighborhoods and combining them with the representation of itself. By denoting H' as the collection of representations in the 1-th layer, its v-th row h/,, i.e., the representation of node e in the 1-th layer, can be obtained as
ZW K ZWK,
V ZWv,
QK
(4)
2 = Attention (ariiomm(,,) K, V) softmax
V,
VF
where We. WK. and Wy FxF denotes trainable projection matrices and F stands for the feature dimen- sions.Further details of GNNs and Transformers are pro- vided in the appendix A.
h = A gggregation ( ((h E (())), h. Combination (h)- ki)),
(1)
To employ a Transformer to graph structure data, is critical to incorporate graph topology appropriately (Zhuo al., 2025). There are two kinds of strategies (Ying et al., 2021; Rampásek et al., 2022). Some methods directly employ adjacency matrix A to regularize the Transformer, e.g., element-wise product A O Sim QK The other methods encode topology structure into position encoding P. e.g., the eigenvector of its Laplacian matrix. Thus, it evident that the utilization of the Transformer
Graph Transformer.
where Aggregation(-) and Combination(,) ) denote the ag- gregation and combination modules, respectively. Follow- ing the above paradigm, classical GNNs, e.g., GCN (Kipf & Welling, 2017)-one of representative GNNs employ the weighted average function AH'-1 to implement the above operations as
GCN: H' - (A'"'-
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Do We Really Need Message Passing in Brain Network Modeling?
for graph-data processing calls for careful handling of graph topology and node representation, consideration that stands out as especially striking in the context of brain network analysis.
90
Classifer
Transformer
Classiter
800
-.0
Scores
Score (%)
70
70-
60
60-
4. Issues with Message Passing
50
AUC
ACC
50-
AUC
ACC
Merric
Metric
This section begins with the method of brain network con- struction, followed by the analysis of the GNNs and Trans- formers on the constructed brain network. Finally, experi- ments are conducted to verify the above analysis.
(a) ABIDE
(b) ADNI
Figure 2. Performances of hasic classifier, GNN and Transformer on ABIDE (left) and ADNT (right) datasets according to AUC and ACC. Note that the simple classifier consistently outperforms basic GNN and Transformer with message- passsngg strategy.
4.1. Brain Network Construction
goes against GNNs' need for different types of informa- tion. Therefore, GNNs' characteristics, especially message passing, can't can't be fully exploited in brain network modeling.
Refer to (Cui et al., 2022) for the detailed construction process of the brain network from raw data. Here, the final two steps, which are closely related to the brain network G, are considered. Firstly, the Brain Region Parcellation segments each subject into ROIs. Secondly, the weights of edges between ROIs are calculated using pairwise Pearson correlation coefficient as
Transformers for Brain Network. As shown in Section 3.3, Transformers learn holistic relations between all tokens with the self-attention mechanism. To employ Transformers in the brain network modeling, the initial token embedding is often set as the adjacency matrix, i.e., X A. Since A is obtained by calculating Pearson correlation coefficients between all ROI pairs, the holistic relationship between all tokens is present in the initial token embedding X. Thus, the necessity of employing Transformers, especially the implicit message passing in the attention mechanism, to explore holistic relations in brain networks is weak (Fig. 1).
E (x. Z (3) 8) Tay DE (2; - 2/2 Ei= (y/ 0)²
(5)
where {Zi,.... } and {yre.... yn are two sequences of response values of two ROIs with the same length n, and i and i are the means of sequences. To obtain robust graph, threshold is often used to sparsify the edge weights
In summary, the paassagg under both GNNs and Transformers can't be fully explored and exploited by con- sidering the characteristics of the brain networks.
Ozz = {
T2gs
if Tay > threshold, 0, otherwise.
The constructed symmetric A = [4xy]] E NNN is seen as the adjacency matrix of the brain network. Note that the sequence (......,,",, } is infrequently characterized as the ....
A=
4.3. Experimental Investigation
To verifyy the above analysis of the ability of GNNs and Transformers in brain networks, this section conducts basic experiments. The performances are measured according to AUC and ACC metrics on ABDIE and ADNI datasets, results of more metrics are placed in Appendix B. Three baselines are involved as follows:
attribute ROI, since its content is the response values at some moments instead of the essential characteristics of the ROI. Section 4.2 elaborates on how GNNs and Transformers construct node attributes.
4.2. Model Analysis
The simple classifier f(A) AW, with A as the feature matrix and W as learnable matrix;
This section analyzes how GNNs and Transformers process brain networks constructed in Section 4.1.
The GNN as in Section 3.2 with A as topology and node attribute;
GNNs for Brain Network. As shown in Section 3.2, GNNs learn representation by combining graph topology and node attributes, which are two different types of information. Unfortunately, the constructed brain network often lacks essential node attributes. To alleviate this issue, existing GNN-based methods often constructed node attributes from the brain network topology, such as (1) identity matrix, (2) Eigenvectors of the adjacency matrix, (3) node degree, (4) local statistic of degree, (5) adjacency matrix itself or from another correlation measurement (Fig. 1). This strategy
The Transformer as in Section 3.3 with as node attribute.
The performances are shown in Fig. 2. It can be observed that the basic and simple classifier consistently outperforms the other two models, i.e., GNN and Transformer. This re- sult meets our analysis in Section 4.2 that message passing under both GNNs and Transformers can't be fully explored and exploited in brain network modeling. Note that some GNN-based and Transformer-bant based models, such as Brain
4

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Do We Really Need Message Passing in Brain Network Modeling?
80
AUCP%)
ACC(%)
SEN(%)
SPE SPE(5) 96)
70
70
68
63
65
70
60
The number layens
number layers
layers
number layers
Figure 3. Performance comparison of QNN (Hadamard product) and GCN (matrix product) on ABIDE dataset according to AUC, ACC, SEN, and SPE. Note that QNN consistently outperforms GCN, both the AUC and the ACC increase the numher of layers increases.
Network Transformer (Kan et al., 2022b), achieve much better performance compared the basic GNN and Trans- former. However, their high performances mainly rely the modification to GNN and Transformer instead of the passing mechanism behind them.
provide further elaboration of Quadratic function and its representative applications in Appendix C.
To verify the effectiveness of the quadratic function with the Hadamard product, the following two brain network encoding methods are compared with Ho = AW as the
initial embedding (0-th layer).
5. Methodology
Graph Convolution Network: H' = 0 O (AH'
Brain Quadratic Network (BQN), which breaks the message- passing scheme, is proposed here.
Quadratic Neural Network: H' H @ (AW"),
where O stands for Hadamard product/element--wiss prod- uct. The performances of the Graph Convolution Network (GCN) and the Quadratic Neural Network (QNN) are shown in Fig. 3. can be observed that the QNN significantly and consistently outperforms the GCN. Besides, both the AUC and the ACC increase as the number of layers increases.
5.1. Motivations
The model analysis and experimental investigation in Sec- tions 4.2 and 4.3 demonstrate that message passing under both GNNs and Transformers can't 't' be fully explored and exploited in brain network modeling. Thus, to enhance the performance of brain network analysis, the fundamental brain network operator should be updated.
In summary, both the expressive ability and universal ap- proximation from theory and experimental results suggest the Quadratic Neural Network (QNN) as an alternative for brain network modeling.
Here, the simple classifier f(A) AW is employed as the base of the investigation for its satisfactory performance. For one-dimension output, f() acts the linear function as f(a) = > w,a, wa, where a [aj..... aNl and [W1 .... WN] denote one-row of A and one-column of W, respectively. It is well-known that linear functions can't implement XOR logic operation (Bishop & Nasrabadi, 2006). Recent attempts (Fan et al., 2018; 2020) demon- strate the expressive ability of the Quadratic/Second-ordee function of the following forms
5.2. Brain Quadratic Network
According to Eq. (7), the Quadratic Network for Brain is formulated by using the Hadamard product as follows.
H' (H' (H'- W/ C (H' (w + (H H W C:
(8)
N
where W and W are the learnable parameters for the 1-th layer. To stabilize the learning process, the first term of Eq. (8) replaced with H o (AW'). which meets the QNN formula in Section 5.1, and the Brain Quadratic Network(BQN) is as follows:
MMa)= wria + by) Σ
Wgal; + (g)
Σ 1=1
wor +
1=1
1=1
-ww,a + b,)( (w))),,.g) W,a by) w. +
where w, = [Wyl, W.N]. Wg = [1g .... J. W. [Wo....., Wo by. by. and are learnable parameters. a² denotes the element-wise square operator, and O stands for Hadamard product product. It is proved that quadratic function can implement XOR logic operation and possesses some better universal approximation properties compared to linear function (Fan et al., 2020; 2025). This motivates us to employ the quadratic function to enhance the performance of brain network analysis. Moreover, we
H' - o (AW%) + (H' O HH' W
(9)
where learnable mappings W and Wi are for original brain network A and the embedding H. As shown in the experiments in Section 5.1, the first term in Eq. (9) plays a key role in performance enhancement, while the second one mainly tends reduce the variance.
5

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Do We Really Need Message Passing in Brain Network Modeling?
5.3. Theoretical Analysis
Wang al., 2016; Luo al., 2022), since its superior per- formance and outstanding interpretability. Therefore, the formula Eq. (10) leads the proposed BQN to seek represen- tation, which can capture mesoscopic community structure to reflect brain functional modules. This provides an inter- pretation of the success of the proposed BQN. In addition to the theoretical analysis, experiments reflecting the cluster- ing characteristics of brain region representations obtained through stacking multiple layers of Eq. (9) are presented in Appendix D.
This section provides deep insight into the Brain Quadratic Network, especially the first term in Eq. (9), i...
H's H' (AW").
(10)
The following theorem provides an intuitive understanding from the concept of community detection.
Theorem 5.1. The iterative formula Eq. (10) of the pro- posed BQN is equivalent to the updating rule of the follow- ing community detection objective function, which is based on nonnegative matrix factorization (NMF) of the adjacency matrix.
6. Evaluations
6.1. Experiments Setup
min WA0.
C(H)- HW A
A
(11)
Datasets. In the experiments, two real-world IMRI datasets are employed: as follows.
where H O denotes all elements in H are non-negative.
Autism Brain Imaging Data Exchange (ABIDE): This dataset is primarily utilized to investigate brain func- tional connectivity variation and structural differences associated with Autism Spectrum Disorder. The pre- processed data version can be accessed from the official website It collects resting-state functional magnetic resonance imaging data from 17 international sites, as well anatomical and phenotypic data. The dataset we used contains 516 Autism Spectrum Disorder patients (ASD) and 493 normal controls (NC).
Proof. The constrained optimization in Eq. (11) can be converted to an unconstrained one by employing a Lagrange multiplier (Bertsekas, 2014). To this end, a Lagrange multi- plier matrix @ is introduced corresponding to the nonneg- ative constraints H 2 0. Thus, the equivalent objective function is
min ((H) = HW] - Al + t(HT)) H
Set derivative of ((H) with respect to H to 0, it obtains
Alzheimer's ' Disease Neuroimaging Initiative (ADNI): ADNI is a widely utilized multimodal neuroimaging repository focused on Alzheimer's disease. The raw images can be obtained from ADNI official website 2 Access is limited and requires adherence request procedure to acquire the data. The dataset for this paper contains 53 Alzheimer's disease (AD) samples and 71 normal controls.
o = 2AW A 2HW WA-
(12)
Following the KKT condition for the non- non-negativity of H, Eq. (12) can be further reformulated as element-wise form
(2AWA- A 2HW A W a), H. = H. 0, (13)
which is the fixed point equation that the solution must satisfy at convergence. Given an initial value of H, the updating rule of H can be formulated as
The construction of brain functional connectivity matrix for ABIDE based on Craddock 200 atlas. The Pearson correlation coefficient between the region-averaged BOLD signals from pairs of ROIs (Regions of Interest) is adopted as the measure of functional connectivity strength between ROIs, namely as brain graph adjacency matrix (Cui et al., 2022; Bessadok et al., 2023; Xu et al., 2024). For ADNI, the fMRI data are first preprocessed using the Data Processing Assistant for Resting-Star -Stte 1MRI (DPARSF) toolkit. Next brain ROIs are defined based on AAL 90 atlas and the average time-series feature is calculated for each individual brain ROI. Pearson correlation coefficient between ROIs is then calculated, which serves the functional connectivity matrix. Specifically, thresholds are set to keep edges with positive weights and drop those with negative weights.
(AW a)is
H Hij (HW) A W A)i3
(14)
where the denominator is to normalize the iterations, and denotes the assignment operation. Therefore, the iteration can be written as in matrix form
H HH @ A).
(15)
U which is the same as the iterative formula of BQN.
The term [HW]- 4 - A in Eq. (11) is the nonnegative matrix factorization of the adjacency matrix A. Nonnega- tive matrix factorization of the adjacency matrix is a widely- used community detection method (Yang & Leskovee, 2013;
org/ abbid/ Pnttps: pestps://addi... //np////niiii///////////... paddi.oooi..... usc.
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Do We Really Need Message Passing in Brain Network Modeling?
Table 1. Performance comparison with two categories of baselines. The best model is bolded and the runner-up is underlined, respectively.
Type
Model
ABIDE
ADNI
AUC1
ACC1
SENT
SPR?
AUC1
ACC1
SENT
SPEt
GCN
59.5943.44
59.30
56.67
61.55
62.45
59.12
54.55
62.00
GAT
60.43
60.10+4.13
59.2645.51
62.89
62.00
58.75+2.86
53.20
64.79
Graph
BrainGNN
64.42
63.09
65.65+22.88
61.81 63.67
1.58
58.72+3.14
52.88+9.70
62.70+-4.26
Neural
BrainGB
3..32+3.66
65.12+3..9
67.01 +:1000
60.07 +8.53
66.44
63.70
8..7
64.67 4:9.07
Networks
FBNETGEN
74.55.13.77
67. 33.09
B11.1119.68
69.61
67. 26
63.2641.36
66.7926.93
61.31
A-GCL
73.8612.91
71.04+2.40
71.42
70.95 19
68. 68.15
67 67.0814.57
68.33 65.45
SAN
71.35:2.18
65.3412.91
55.41
66.11 67.39
61.7814..2
53.94 56
63.63
Graphormer
63.91144.0
61.88
66.30
55.74111.00
60.69
60.18 55.7513311
47.75= 13.53
Graph
Grans
60.13:26.3
47.83
19.77 10..0
:3.65
58.60.5.41
65.57 0.
54..77 :3.42
Transformer
BrainNETTF
77.93+1.41
69.26+2.26
65.92+ ...
73.20
69.73#2.61
67.85 67.85+2.92
63.61
70.67 +8.33
Models
ContrastPool
57.3640.87
57. 57.4440.69
57.66 57.66446.86
57.08
68.17
66.21
61.5147.44
72.43
ALTER
77.99
70.10+2.26
72.8447.40
67. 67.68
71.86
66.92+3399
71.55
64.00++.88
BioBGT
69.9641.18
69.70
67. 07.0444..4
72.02
63. 63.1643.74
62.27+3.23
63.97
60.55
BQN (Ours)
79.85 42.53
73. 73.26 26
72.03
74.18
68.62 62.22...22
70.91
68. 68.00
Baselines. Thirteen Baselines are compared in the experi- ments, which can be divided into two categories.
epochs is set to 200 with batch size 16. An adam optimizer is adopted with initial learning rate of 10-4 and weight de- cay 10 while training, target learning rate is from 10 to 10-1, The activation function we selected is LeakyRel.U. The number of layers, i.e., k is selected from 1, 2, 3, 4, 5 and the dropout rate is chosen from 0., 0.1, 0.2, 0.3. The results were averaged over random runs. Notably, existing studies of GNNs and Transformers for brain networks show marked differences in data processing and model selection, which poses challenges in fairly evaluating their performance. To address this, this paper unifies these settings and compares methods in fair manner. The details of corresponding unified settings are elaborated in the Appendix F.
Graph Neural Network (GNN)-based models, includ- ing two typical GNNs: GCN (Kipf & Welling, 2017) and GAT (Veličković et al., 201 2018)), and four brain- specific GNNs: BrainGNN (Li et al., 2021), BrainGB (Cui et al., 2022), FBNETGEN (Kan et al., 2022a) and A-GCL (Zhang et al., 2023).
Graph Transformer (GT)-based models, including three typical GTs: SAN (Kreuzer et al., 2021), Graphormer (Ying et al., 2021) and GraphTrans (Wu et al., 2021a)), and four brain-specific GTs: Brain- NETTF (Kan et al., 2022b), ContrastPool (Xu et al., 2024), ALTER (Yu et al., 2024) and BioBGT (Peng et al., 2025).
6.2. Result Analysis
Brain Disorder Disease Classification. The performance comparison between BQN and the baseline models on the ABIDE and ADNI datasets are presented in Tab. 1. Upon observation, is evident that compared to GNN-based and GT-based baselines, BQN exhibits superior performance in terms of AUC and ACC metrics. In particular, on the ABIDE dataset, BQN achieves an accuracy that is 2.43% higher than that of the runner-up baseline ALTER. This advantage can be attributed to the fact that BQN directly leverages the essential information from brain functional connectivity matrices. Therefore, reduces model complex- ity and inference uncertainty that may arise from extraneous factors. Moreover, during training, BQN implicitly learns characteristic representations of brain functional modules, as evidenced in Sec. 5.3. These learned representations of functional regions enhance the accuracy of brain disorder classification. By directly modeling the intrinsic proper- ties of brain functional connectivity, BQN is particularly well-suited for brain disorder classification tasks, especially
Additional comparative experiments of BQN are conducted on supplemental datasets, please refer to Appendix E.
Metrics. To conduct comprehensive evaluations of per- formance, a combination of machine learning and med- ical diagnostic-specific metries are employed, including Area Under the Receiver Operating Characteristic Curve (AUC), Accuracy (ACC), Sensitivity (SEN), and Specificity measure of
(SPE). AUC provides a parrshold-indepeneee the model's ss ability to distinguish between classes. ACC is straightforward metric to assess the model's 's classification performance. SEN, also known as the true positive rate, is a critical metric for medical diagnostics. In contrast, SPE represents the true negative rate.
Implementation Details. The experiments are performed on GeForee RTX3090 GPU. For all datasets, we employ random splits with the ratio 7:1:2 to get the training set, val- idation set and test set. Furthermore, the number of training
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Do We Really Need Message Passing in Brain Network Modeling?
ABIDE
ADNI
Table Comparison of running time(s) on ABIDE and ADNI.
90-
90
wib naintel
wiomoshal
Method
ABIDE
ADNI
Score:
80
Score: (%)
GCN
14.26
4.12
70
FBNETGEN
19.73
5.71
60
g
A-GCL
41.57
6.42
50
ACC
ACC
SEN
SPE
50
AUC
ACC
SPE
SAN
712.01
152.23
Metric
(b) Metric
Graphormer
973.52
179.11
BrainNETTF
17.32
4.77
Figure 4. Performance comparison of BQN with and without
ContrastPool
283.02
76.05
Hadamard residual term on two datasets.
ALTER
37.21
7.12
BioBGT 31.24
6.78
84
76
BQN(Ours)
11.31
3.33
80
72
AUC(%)
76
ACC(%)
when dealing with the complex and limited-scale nature of neuroimaging datasets. These experimental results and analyses indicate superior performance and highlight the promising potential of BQN.
72
68
ARIDE
64
ABIDE
64
ADNI
ADNI
60
60
The number of layers
(b) The number of layers
Effficiency Test. This experiment aims to thoroughly amine the efficiency of the proposed BQN. It achieves this by comparing the running time of BQN with those of nine representative baselines. The times reported in Tab. 2 are the total times of training 100 epochs in full batch mode. It can be observed that compared to all baselines, the proposed BQN achieves the shortest runtime on ABIDE and ADNI datasets. This is mainly attributed the simplicity of the proposed BQN, which guarantees its efficiency. To be spe- cific, the core computation of BQN involves element-wise multiplication of matrices via Hadamard product, which poses à quadratic complexity of On²)), where n denotes ROIS number. In contrast, both GNN-based and GT-based models require the computation of matrix multiplication between two square matrices (the propagation matrix and the feature matrix, e.g., Pearson matrices), which typically involves cubic complexity of O(n²).
Figure 5. Performance variations for varying & on two datasets.
Fig. 5. From this figure, it is evident that the proposed BQN achieves stable performance on these two datasets when the parameter & is within the range {1, 2, 3]. This demonstrates the stability of BQN. Moreover, the experimental results indicate that the model performance exhibits gradual de- cline as the number of layers increases. Considering that brain datasets are generally of limited size (on the order of hundreds of samples), the observed performance degrada- tion can be attributed to overfitting, which likely induced by the excessive number of parameters and the heightened model complexity associated with deeper architectures.
Case Study. To interpret the rationality of BQN on the ABIDE dataset, we construct two contrast brain graphs that reflect brain network differences among distinct groups The first contrast brain graph is derived from the initial brain connectivity matrices, while another contrast brain graph is from the brain network generated by BQN. Specifically, the brain connectivity matrix template of ASD is obtained by averaging functional connectivity matrices within ASD groups, *Tenglate E AASD Brain template of NC is obtained similarly. Tomplate m denotes the number of normal subjects, and n represents the number of ASD patients in the ABIDE dataset. The contrast brain graph is then calculated as: Acontrat A NC templatc O Template where @ performs element-wise subtraction on the two input matrices. The learned contrast brain graph is then obtained by performing the same process as the equations above. The only difference is that the output of the last layer of well-trained BQN is as brain graphs to construct the learnable contrast graph. Notably, the input of the well-trained BQN is a test set, and the phenotype of the subject is based on the result of the classification head of the model.
Additional Experiments w/o Hadamard residual. To val- idate the rationality and effectiveness of residuals, i.e., the second term proposed by Eq. (9). An ablation study is conducted with an architectural theme defined as H() H(111) AWi with or without (H'-1 H' W mod- uel, where H = AW, W represents a linear layer. The comparison results are shown in Fig. 4. It can be seen that for the AUC, ACC, and SEN metrics, the addition of the Hadamard residual provides consistent performance improvements on both ABIDE and ADNI datasets. This il- lustrates the validity of the Hadamard residual. Besides, for AUC and ACC metrics, the standard deviations of the model results are smaller, indicating that the Hadamard residual also contributes to the stability of the model.
ANC
1 Σ ANC where m
A ASD
Hy plperpaareeetr Sensitivity Analysis. This experiment is designed to offer an intuitive understanding of how to select the optimal number of layers k. The effect of this hyperparameter on the model performance is reported in
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Do We Really Need Message Passing in Brain Network Modeling?
BQN consistently outperforms models based GNNs and Transformers. We hope this study could offer valuable in- sights into simpler models that can also achieve remarkable performance in brain disorder disease classification.
R
R
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (No. U22B2036, 62376088, 62025604, 92370111, 62272340, 62261 136549), in part by the Hebei Natural Science Foundation (No. F2024202047), in part by the National Science Fund for Distinguished Young Scholarship (No. 62025602), in part by the Hebei Yanzhao Golden Platform Talent Gathering Programme Core Talent Project (Education Platform) (HI/ZD202509), in part by the Post-graduate's 's Innovation Fund Project of Hebei Province (CXZZBS2025036), in part by the Tencent Foundation, and in part by the XPLORER PRIZE.
(a) Contrast brain graph constructed from initial data
R
(b) Contrast brain graph constructed from the output of BQN
Impact Statement
Figure 6. Brain network comparison. Red edges denote significant differences corresponding brain connectivity between the nor- mal population and individuals with Autism Spectrum Disorder.
This paper presents work whose goal is to advance the field of Machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
The visualization of the two contrast graphs, the top-20 edges with the largest weights are selected, is shown in Fig. 6. Firstly, it can be observed that compared to the initial contrast brain graph (subfigure a), the contrast graph learned by BQN is sparser and retains the core focus of abnormal connectivity, such as connectivity between paracentral lob- ule region and cingulum region. This finding indicates that the BQN learn more holistic brain functional connectivity and gain ability of learning characteristic representations of brain functional modules, consistent with the theoretical analysis.
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Do We Really Need Message Passing in Brain Network Modeling?
In the appendix, the presented contents follows:
A: Further elaboration on GNNs and Transformers.
B: More metrics' analysis.
C: Theoretical and application progress of quadratic networks.
D: Experiments the clustering characteristics of node representations.
E: Further comparative experiments of BQN.
F: Uniform configurations.
A. Further Elaboration on GNNs and Transformers
To better illustrate the message passing mechanism of GNNs, we adopt GCN, SGC (Wu et al., 2019), and APPNP (Klicpera et al., 2019) as examples. These models employ the weighted average function to implement Eq. (1) and Eq. (2) as:
GCN: H' = OAHH'
SGC: H' ÁH' A'x,
(16)
APPNP: H' = -1-a) ( + 4aX,
where denotes the non-linear activation function (c.g., ReLU(-)), is the balancing hyper-parameter and H X denotes the initial node attributes. Á (D I,) IDD I,) represents the normalized adjacency matrix with self-loop. Above three models achieve enhanced neighborhood aggregation and residual combination by employing meticulously designed message passing function, which significantly improves the model's performance and accuracy in capturing complex patterns and relationships within the data. Recently, GNNs struggle to capture long-ram dependencies since stacking multiple layers tends to cause over-smoothing and over-squashing issue (Li et al., 2018; Topping et al., 2022) and leads to the loss of discriminative information.
For the representation learning function Eq. (4) Transformer Section 3.3, can be reformulated in token-wise form as
Vv Σ exp(sim(qu. k.))
2
Σ SQK NEV
nEV Evey exp(simm(s. (q,, k.))
Vas
(17)
where sim(.) terms the similarity function, which mainly adopts scaled dot-product attention, i.e., sim Q) OKT/VF: Thus, the attention score matrix SQK E R acts as the propagation weights, and the Transformer follows the holistic message-passingg paaggggggpsinggggge scheme.
B. More Metrics' Analysis
Owing to the restricted number of datasets employed and the absence of consistent trends across multiple metrics, we conducted additional experiments to support the conclusions as presented in Fig. 2, 3 and
To evaluate the performance of the basic classifier, GNN and Transformer (Fig. 2), we further conducted experiments on the ABIDE and ADNI datasets, incorporating Precision and Recall as metrics. The results are presented in Fig. 7. These additional experiments reveal consistent trends across the datasets, supporting the hypothesis that the simple classifier outperforms the basic GNN and Transformer with cesssage-pssinng strategy.
For Fig. 3 (performance comparison of QNN and GCN) and Fig. 5 (performance variations for varying layers), we adopted the micro-Fl score to provide a more comprehensive evaluation of model performance in the above two experiments. The experimental results are presented Fig. 8. Subfigure (a) shows that the QNN significantly and consistently outperforms the GCN as the number of layers increases. subfigure (b), BQN achieves stable performance when the number of layers increases, with the best performance occurring at layer 3 on the ABIDE dataset. In contrast, the results indicate gradual performance decline the number of layers increases the ADNI dataset. These experimental trends align with the findings from the main-bodk body experiments in Fig. 3 and 5.
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75
Clsssifer
Classifier
Transformar
70
ONN
70
Score: (%)
70
Score: (%)
iitaro Fal..FFle ((%)
68
micro-F F1%))
66
65
60
66
82
60
64
55
62
58
ARIDE
ADNI
50
Precision
Recall
50
Precision
Recall
Metric
Metric
The number of layers
The number of layors
(a) ABIDE
(b) ADNI
(a) QNN vs. GCN
(b) Sensitivity of layers.
Figure 7. Performances of basic classifier, GNN and Transformer on ABIDE (left) and ADNI (right) datascts according Precision and Recall. Note that the simple classifier consistently outperforms basic GNN and Transformer with message-passiine paaigggg-annggggg strategy.
Figure Performance comparison of QNN (Hadamard product) and GCN (matrix product) on ABIDE dataset according micro- Fl (left), and performance variations for varying k on ABIDE and ADNI datasets according micro-F1 (right).
C. Theoretical and Application Progress of Quadratic Networks
The Quadratic Neural Network (QNN) was proposed by authors in (Fan et al., 2018). Subsequent researches have further explored the expressive capabilities of quadratic networks. For instance, (Fan et al., 2020) presented four theorems highlighting the ad advantages of QNNs over conventional networks in expressive efficiency, unique representation, compact architecture and computational capacity. More recently, (Fan et al., 2025) demonstrated that QNN can implement XOR logic operations and exhibit superior universal approximation properties compared to linear functions.
The findings mentioned above have inspired numerous application-orienttd studies. AFT (Zhai et al., 2021) introduces an efficient alternative quadratic function traditional Transformers that eliminates the need for dot product self-attention, demonstrating competitive performance on all the benchmarks, while providing excellent efficiency at the same time. QCNN (Liao et al., 2023) proposes novel quadratic convolutional neural network for bearing fault diagnosis, incorporating a derived attention mechanism termed qttention that enhances interpretability and effectiveness. QNN-Bi-LSTM (Keshun et al., 2024) utilize a hybrid model combining a quadratic neural network (QNN) with a bidirectional long short-term memory network (Bi-LSTM) for efficient and interpretable rolling bearing fault diagnosis. Graph Reciprocal Network (GRN) (Yang al., 2023) treats node attributes and topology of graph as reciprocal elements by regarding nodes as another kind of attribute and employing novel representation scheme based on Quadratic Networks achieve fine- tne-grained element-wise product representations.
Previous researches have shown the broad applicability of Quadratic Networks across various domains. However, no studies have yet applied Quadratic Networks-based methods to brain network analysis. Our work fills this gap by showing that Eq. (9) of the quadratic function reveals a clustering effect in brain functional regions. The competitive performance of the Brain Quadratic Network (BQN) presented in Section 6 underscores the potential of Quadratic Networks for brain network analysis and validates its reasonable application in this context.
Table 3. Clustering performance with increasing numher of layers
Metric
init layer_1 layer layer_3
SCt
0.004
0.179
0.250
0.327
CHt
5.746
12.357
18.877
22.528
DB.
4.974
3.434
2.674
1.795
D. Experiments on the Clustering Characteristics of Node Representations
The node representation obtained through iterative learning of the multi-layers of Eq. (9), its clustering characteristic has been proved in Section 5.2 theoretically. In this section, we further experimentally validate its clustering property. Following the segmentation criteria outlined in (Dosenbach et al., 2007) and (Dosenbach et al., 2010), the brain is segmented into corresponding functional regions. To assess the clustering characteristics of BQN, we employ three standard cluster metrics: the Silhouette Coefficient (SC), both the original data and the outputs of three well trained BQN models, using layer, 2 layers and layers respectively,
Index (CH) and Davies-Bine Binould Index (DB). Results are obtained from
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which are presented in Tab. 3. It can be observed from the table that the proposed BQN effectively captures the clustering properties of functional brain regions. Notably, the results reveal that clustering performance increases as the number of layers increase when the number of BQN layers is within {1, 2,, 3}. which is consistent with Section 5.3.
E. Further Comparative Experiments of BQN
Considering the limited number of datasets and related studies used in main body, further comparative experiments are employed on ADHD-200 and PPMI datasets. The details of the two datasets and studies we selected are follows:
Datasets. (1) Attention Deficit Hyperactivity Disorder AADHD-200) dataset is collaboration of 8 international imaging sites that has aggregated and openly sharing neuroimaging data from children and adolescents diagnosed with ADHD and typically developing controls. The dataset used in this paper includes 459 subjects, comprising 230 typically normal individuals and 229 ADHD patients. (2) Parkinson's 's Progression Markers Initiative (PPMI) dataset aims to identify biological markers of Parkinson's 's risk, onset and progression. Similar to ABIDE, ADNI and ADHD-200 200, PPMI is also a multi-site dataset. For this paper; the dataset includes 15 normal controls (NC), 67 prodromal individuals, and 113 Parkinson's disease (PD) patients.
The ROI definition in ADHD-200 200 dataset is based on Craddock 200 atlas, brain graphs are constructed by computing the Pearson correlation coefficients. Specifically, pearson matrices serve as node attributes and brain functional connectivity matrices. For PPMI dataset, ROI definition is based on AAL 1 116 atlas and node features, edge weights were preprocessed by authors in (Xu et al., 2023).
Baseline Methods. Three supplemental baselines are appended to comparison experiments, including A-GCL (Zhang et al., 2023), AGT (Cho al., 2024) and BrainMGT (Shehzad al., 2025). Furthermore, the ALTER employed in the primary text utilized a baseline for this comparison experiment, thereby providing stronger contrast. The first two studies are GNNs specially designed for Brain Network Analysis, while BrainMGT and ALTER is based Transformer.
Metrics. AUC, ACC, SEN and SPE are used to evaluate models performance on ADHD 200 dataset, consistent with the ABIDE and ADNI datasets. For PPMI dataset, we utilize Accuracy, Precision, Recall and Specificity to assess the classification performance of all basclines. Besides, model efficiency comparisons, c.g., training -imes),, are included on both two datasets.
Implementation Details. For the above two datasets, random partitions were used with ratio of 7 : The training process was configured for 200 epochs, with the batch size within the range of {4,8, 8, 16). Notably, to ensure fair assessment of the baselines in conjunction with BQN, the epoch exhibiting the lowest loss on the validation set is selected for performance evaluation on the test set across all models. The Adam optimizer, number of layers, and dropout rate were consistent with the configuration selection used for the ABIDE and ADNI datasets in Section 6.1. Moreover, training times reported in Tab. are the total times of training 100 epochs in full batch mode on each dataset.
Experimental Results. The performance comparison between BQN and the baseline models on the ADHD-200 and PPMI datasets are presented in Tab. 4. Note that AGT is specifically developed for multi-class classification tasks, whereas ADHD-200 200 is binary classification task. Thus, performance comparison with AGT is limited on PPMI dataset, which has three classes. The results reported in the table indicate that BQN achieved optimal performance in three of the four classification metrics on the ADHD-200 dataset, demonstrating its superiority. On the PPMI dataset, BQN is not superior to AGT but still shows competitive performance. Furthermore, in terms of model efficiency, our approach exhibits the shortest training time across both datasets. Above experimental results highlight the promising potential of our model.
Table Performance comparison on ADHD-200 and PPMI. The best and the runner-up models is in bold and underlined, respectively.
Model
ADHD-200
PPMI
AUCt
ACCt
SENt
SPEt
Time(s))
Accuracyt
Precision?
Recallf
Specificity? Time(s))
AGT
77.69 77.6942.81
56.42
73.69
79.37
11.08
A-GCL
74.78.14.9
73. 44.30
72.01
73.08
19.71
73.6543.19
48.9943.71
70.7323.01
75.67=2.19
14.12
ALTER 83.1641.61
73.4841134
71.88
52.20
17.38
73.9143.18
50.29
21.81
77.36-4.22
11.55
BrainMGT
78.3341.10
71.2742.26
71. 71.09
73. 43.4
26.35
70.61
47.10+2226
68. 68.4524.81
74.29_a.40
17.55
BQN (Ours) 83.3411.13
75.68 1.05
79. 7.27
71. 71.63
6.56
71.2613.95
52.30
73.9324.22
76.25
4.79
14

---

Do We Really Need Message Passing in Brain Network Modeling?
F. Uniform Configurations
Existing GNNs and Transformers for brain networks are very different data processing and model selection, which makes it difficult fairly assess their performance. To alleviate this difficulty, we unify these settings and compare methods in fair manner. Unifying settings our work can be concluded two aspects:
Model Selection, We unify early stopping criteria as the lowest loss on the validation set, which is consistent with BrainGNN and many traditional GNNs and Graph transformers.
Data Processing. We employed uniform preprocessing approach and brain network construction method for all models on each dataset. Specifically, for the ABIDE dataset, our preprocessing and brain network construction were aligned with those used by BrainNETTF, ALTER and BioBGT. While for the ADNI dataset, we maintained consistency with ALTER's preprocessing and brain network construction methods. For the ADHD-200 dataset, our preprocessing and brain network construction were consistent with BioBGT's methods. Finally for the PPMI dataset, the node attributes and edge weights were preprocessed by authors in (Xu et al., 2023).
15


Reviews:
[{"review_id": "2HwOzRfaFM", "reviewer": "Reviewer_TdBX", "summary": "This paper investigates the GNN and Transformer, which follows the message passing pipeline, in brain network modeling. It observes that these two methods can’t enhance the performance compared to the vanilla classifier. Following by the analysis of the weakness of them from the brain network construction, it presents a novel and simple method based on Quadratic Network, i.e., Hadamard product, which has attractive properties, such as efficiency and latent clustering. Experiments verify the statement and the superiority of the proposed methods.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. The model analysis on GNN and transformer provides evidence to question the message passing in brain modeling. The proposed BQN makes sense by theoretical analysis on its clustering property. Its rationality is also guaranteed by the theory progress in the Quadratic Network. I suggest that the authors include them in the manuscript and emphasizing model efficiency in the introduction and abstract.\n\nR1. Thanks for your suggestion. We will incorporate the mentioned contents of the Quadratic Network into the manuscript, referencing [1], [2], [3], and [4]. Additionally, we will emphasize the model efficiency in both the introduction and abstract to better highlight the contributions of this paper.\n\n[1]Universal approximation with quadratic deep networks. Neural Networks, 2020\n\n[2]An attention free transformer. arXiv preprint, 2021\n\n[3]Attention-embedded quadratic network (qttention) for effective and interpretable bearing fault diagnosis. IEEE Transactions on Instrumentation and Measurement, 2023\n\n[4]Towards efficient and interpretative rolling bearing fault diagnosis via quadratic neural network With Bi-LSTM. IEEE Internet of Things Journal, 2024.\n\n> Q2. The source code is not provided. I suggest including it in the supplementary material.\n\nR2. According to your suggestion, the source code has been made available at [https://anonymous.4open.science/r/BQN-demo](https://anonymous.4open.science/r/BQN-demo) for verification purposes.\n\n> Q3. The experimental evidence should be provided to justify its clustering property.\n\nR3. In response to your suggestion, we have analyzed the clustering properties of the proposed BQN model. The brain is segmented into functional regions using established criteria from prior research [5][6]. The clustering performance is evaluated using three standard metrics: Silhouette Coefficient (SC), Calinski-Harabasz Index (CH), and Davies-Bouldin Index (DB). Results are obtained for both the original data and the outputs of three well trained BQN models, using 1 layer, 2 layers and 3 layers respectively.\n\n|     | init | layer_1 | layer_2 | layer_3 |\n| --- | --- | --- | --- | --- |\n| SC↑ | 0.004 | 0.179 | 0.250 | 0.327 |\n| CH↑ | 5.746 | 12.3536 | 18.877 | 22.528 |\n| DB↓ | 4.974 | 3.434 | 2.674 | 1.795 |\n\nThe results reveal that the proposed BQN effectively captures the clustering properties of functional brain regions.\n\n[5]Distinct brain networks for adaptive and stable task control in humans. Proceedings of the National Academy of Sciences, 2007\n\n[6]Prediction of individual brain maturity using fMRI. Science, 2010\n\n> Q4. There are some methods based on prototypes, such as [Kan et al., 2022b], the essence of which is clustering. Why does BQN outperform them?\n\nR4. The performance superiority of the proposed BQN is primarily due to its quadratic network’s approximation capabilities and robust model architecture.\n\nFirstly, BrainNETTF[Kan et al., 2022b] use the Pearson matrix as the feature matrix, which already captures holistic brain region interactions. This limits the capacity of the Transformer's fully connected graph messaging mechanism. In contrast, BQN learns clustering properties and employs a quadratic network with more general function approximation capabilities, resulting in better ROI and graph representations. Secondly, BrainNETTF is a Transformer-based model with larger parameters, increasing the risk of overfitting on relative small brain datasets. BQN, with fewer parameters, reduces model complexity and inference uncertainty.", "claims_and_evidence": "I appreciate that this paper can explore this essential question. It ignores existing methods’ modifications to GNN and Transformer and only shows the abilities of GNN and Transformer. \nThe claims are supported by clear and convincing evidence:\nBoth model analysis and experiments demonstrate that message passing is not necessary in brain modeling.\nBoth theoretical analysis and experiments verify the property of the proposed BQN.", "methods_and_evaluation_criteria": "The model analysis on GNN and transformer provides evidence to question the message passing in brain modeling. The proposed BQN makes sense by theoretical analysis on its clustering property. Its rationality is also guaranteed by the theory progress in the Quadratic Network. I suggest that the authors include them in the manuscript.", "theoretical_claims": "I checked the theorem and its proof.", "experimental_designs_or_analyses": "The design of experiments on both motivations and evaluation of proposed methods are convincing since the experimental setups are common in this field, including datasets, baselines, and criteria.", "relation_to_broader_scientific_literature": "Most SOTA methods on brain network modeling are based on GNN and Transformer, which follow the message passing mechanism. This paper questions its necessity and presents a simple, efficient, and novel method. I believe it may significantly impact this field. I also think it will motivate us to consider whether GNNs are necessary in many fields.", "essential_references_not_discussed": "Sufficient.", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "The source code is not provided. I suggest including it in the supplementary material.\nEfficiency is a remarkable property of the proposed BQN compared to methods based on GNN and Transformer. However, it is only verified in the experiments. I suggest emphasizing it in the introduction and abstract. \nThe experimental evidence should be provided to justify its clustering property.", "questions_for_authors": "In addition to the points in Other Comments or Suggestions, I have another concern. There are some methods based on prototypes, such as [Kan et al., 2022b], the essence of which is clustering. Why does BQN outperform them?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wbgfRufopB", "reviewer": "Reviewer_fMGi", "summary": "The paper proposes Brain Quadratic Network (BQN), a novel approach for brain network modeling that replaces traditional message-passing mechanisms with quadratic networks and Hadamard products. It shows that BQN outperforms GNNs and Transformers on fMRI datasets, achieving higher accuracy and efficiency. Theoretical analysis reveals that BQN implicitly performs community detection, capturing brain functional modules.\n\n## update after rebuttal", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1.The parameters *b* and *c* in Eq. 7 are not represented or utilized in BQN(Eq. 8). Should the layers of the model use MLP?\n\nR1. Yes, the matrix $\\mathbf{W}$ in Eq. 8 represents a Multi-Layer Perceptron (MLP), and the parameters $b$ and $c$ in Eq. 7 are realized in the model implementation by setting bias=True for the MLPs.\n\n> Q2.The results are somewhat insufficient. Given the limited number of datasets used and the lack of consistent trends across multiple metrics, it is recommended that the authors provide additional experimental results for other metrics in Figures 2, 3, and 5.\n\nR2.According to your suggestion, we have conducted additional experiments to supplement the results in Figures 2, 3, and 5. For Figures 2 and 3, we have incorporated precision, recall and micro-F1 metrics on experiments with the ABIDE and ADNI datasets. The results are available at the following links: [Fig2_ABIDE](https://anonymous.4open.science/r/BQN-demo/figure/Fig2_ABIDE.jpg), [Fig2_ADNI](https://anonymous.4open.science/r/BQN-demo/figure/Fig2_ADNI.jpg), [Fig3_F1](https://anonymous.4open.science/r/BQN-demo/figure/Fig3_F1.jpg). These supplementary experiments demonstrate consistent trends across the datasets. For Figure 5, we utilze the micro-F1 score metric to provide a more comprehensive evaluation of model performance as the number of layers increases. The results can be accessed at [Fig5_F1](https://anonymous.4open.science/r/BQN-demo/figure/Fig5_F1.jpg).\nWe believe these additional results enhance the robustness of our findings and address the concerns regarding the sufficiency of the experimental evidence.\n\n> Q3.Some punctuations are missing after certain formulas, such as Equations 6 and 8. In Equation 8, it seems that *$W_A$* is reused\n\nR3. Thank you for pointing this out. We will carefully review the manuscript to ensure proper punctuation is used throughout, especially after equations. We will revise Equations 6 to:\n\n$$ \na\\_{xy} = \n\\begin{cases} \nr\\_{xy}, & \\text{if } r\\_{xy} > \\text{threshold}, \\\\\\\\\n0, & \\text{otherwise}.\n\\end{cases} \n$$\n\nAnd for Equation 8, we will revise it to:\n\n$$\n\\mathbf{H}^l = (\\mathbf{H}^{l-1} \\mathbf{W}_A^l) \\odot (\\mathbf{H}^{l-1} \\mathbf{W}_B^l) + (\\mathbf{H}^{l-1} \\odot \\mathbf{H}^{l-1}) \\mathbf{W}_C^l,\n$$\n\nFor $\\mathbf{W}_A^l$ is reused, we use $\\mathbf{W}_A^l$, $\\mathbf{W}_B^l$ and $\\mathbf{W}_C^l$ instead. And we will check variable symbols throughly to correct any reuse errors.", "claims_and_evidence": "Yes, the claims are well-supported by both theoretical analysis and extensive experiments. Theoretical connections to community detection via non-negative matrix factorization (NMF) validate BQN's ability to capture brain network structures. Empirical results on graph datasets demonstrate superior performance and efficiency compared to GNNs and Transformers.", "methods_and_evaluation_criteria": "Yes, the proposed Brain Quadratic Network (BQN) and the evaluation criteria using fMRI datasets are appropriate for the problem of brain network modeling.", "theoretical_claims": "Yes, I checked the correctness of the proof for Theorem 5.1, which connects the Brain Quadratic Network (BQN) to nonnegative matrix factorization (NMF) for community detection.", "experimental_designs_or_analyses": "Yes, I checked the soundness of the experimental designs and analyses. The experiments on the ABIDE and ADNI datasets are well-designed, and the results are valid. The comparisons with GNN and Transformer baselines are appropriate, and the performance metrics used are suitable for evaluating the classification tasks.", "relation_to_broader_scientific_literature": "The paper's contributions are well-aligned with the broader literature by extending quadratic networks to brain network modeling, connecting the model to community detection via NMF, and providing a simpler, more efficient alternative to GNNs and Transformers.", "essential_references_not_discussed": "No, the paper does not omit any essential related works.", "other_strengths_and_weaknesses": "**Strengths**\n\n1) The paper is well-written and easy to follow, effectively conveying its contributions.\n\n2) The paper provides experimental results on benchmark datasets.\n\n**Weaknesses**\n\n1) The parameters $b$ and $c$ in Eq. 7 are not represented or utilized in BQN (Eq. 8). Should the layers of the model use MLP?\n\n2) The results are somewhat insufficient. Given the limited number of datasets used and the lack of consistent trends across multiple metrics, it is recommended that the authors provide additional experimental results for other metrics in Figures 2, 3, and 5.", "other_comments_or_suggestions": "Some punctuations are missing after certain formulas, such as Equations 6 and 8.\nIn Equation 8, it seems that $W_A$is reused.", "questions_for_authors": "See Weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FoggWKupoa", "reviewer": "Reviewer_nwPW", "summary": "The paper identifies a limitation in the message-passing framework for brain network analysis and proposes an approach, the Brain Quadratic Network (BQN), to address this issue. BQN demonstrates superior performance compared to standard Graph Neural Networks (GNNs) and graph transformers on widely used brain network datasets, highlighting its potential for advancing graph-based neuroimaging analysis.\n\nUpdate of review after rebuttal: the authors have addressed my concerns and I have increased my rating.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. Stacking multiple layers of Eq. 10 is theoretically equivalent to a single-layer formulation. The paper lacks theoretical or empirical justification for stacking multi-layer of Eq. 10 leads to performance gains.\n\nR1. There may be a serious misunderstanding. Stacking multiple layers of Eq. 10 is **NOT** equivalent to a single-layer formulation. Take the case of a two-layer BQN as an example. Formulation $AW_A^1 \\odot AW_A^2$ at the vector level is equivalent to $(aw_a^1) (aw_a^2)$, where $W_A^l$ means linear transformation of layer $l$. Subsequently, take the example of $a$ contains two variables, $(aw_a^1) (aw_a^2) = (a_1w_{a1}^1 + a_2w_{a2}^1) (a_1w_{a1}^2 + a_2w_{a2}^2)≠(a_1w_{1} + a_2w_{2})$ is clearly obtained for multiply interaction of variables $a_1$ and $a_2$. Thus, at the matrix level $AW_A^1 \\odot AW_A^2≠AW_A$, which means stacking multiple layers of Eq. 10 is not equal to the single-layer formulation.\n\nBesides, the performance gains of multi-layer come from the captured community structure in Eq. 11, which is the objective function of NMF-based community detection. Eq. 14 is the **iterative** updating rule of solving Eq. 11, and is equivalent to Eq. 10. Thus, multi-layers of Eq. 10 is the **iterative** updating rule of solving Eq. 11, which can learn better community structure and guarantee the performance gains.\n\n> Q2. More related studies and large datasets should be included.\n\nR2. We have incorporated the mentioned methods into our effectiveness and efficiency comparison and expanded our evaluation to include two additional datasets (ADHD-200 and PPMI), more large-scale than ADNI. All methods use the same data preprocessing, brain network construction and uniform data partition.\n\nNo comparison was made with the work [3] for no open source. AGT requires datasets with more than two classes, limiting its evaluation on the PPMI dataset. The experimental results are shown in https://anonymous.4open.science/r/BQN-demo/figure/Table.jpg\n\nTable reveals that BQN outperforms the baselines on the ABIDE, ADNI, and ADHD-200, demonstrating its superiority. On the PPMI dataset, BQN is not superior to AGT but still shows competitive performance while achieving the best efficiency. This highlights the promising potential of our model.\n\n> Q3. The reported results of ALTER and ContrastPool are different from original papers. ?\n\nR3. Existing GNNs and Transformers for brain networks are very different in data processing and model selection. It makes it difficult to fairly assess their performance. To alleviate this difficulty, **this paper unifies these settings and compares methods in a fair manner**.\n\n- **model selection**. We unify early stopping criteria as the lowest loss on the validation set. This is different from ALTER, which uses the criteria of the highest AUC. This causes the performance gap of ALTER. We also find that the proposed BQN also outperforms ALTER with the criteria of highest AUC.\n- **data processing**. We unify data preprocessing and brain network construction as BioBGT[ICLR'25], ALTER, and BrainNETTF. This is different from the ContrastPool, which follows [5]. This causes the performance gap of ContrastPool. Existing work has demonstrated that different brain network construction methods can lead to different performance.\n\n[5]Data-driven network neuroscience: On data collection and benchmark. NeurIPS, 2023\n\n> Q4. A convergence plot is expected to support a reliable trained model using the ADNI dataset.\n\nR4. The convergence plot, which includes training loss, validation loss, and test accuracy using BQN, is available at https://anonymous.4open.science/r/BQN-demo/figure/BQN_convergence.jpg. The overall trend of the three experimental metrics is steady, despite some fluctuations. This indicate that the model generalizes well without severe overfitting.\n\n> Q5. The authors should clarify the chosen readout function theoretically aligns with BQN framework.\n\nR5. We employ OCREAD, which concatenates the embeddings of cluster centers, as the readout function in our proposed method, consistent with BrainNETTF and ALTER. OCREAD exploits the clustering property for readout, which is consistent with the quadratic network in BQN as shown in Theorem 5.1.\n\nBesides, traditional readout mechanisms, such as MEAN and MAX, remain valid for BQN, since they meet the requirements of readout mechanisms from node embeddings to graph embeddings. However, we believe their performance is not as good as OCREAD, since the clustering property is not considered.\n\n> Q6. Elaborate the transformation from the output of the model to brain graphs and provide thresholding details in the conversion.\n\nR6. Given the output of the final BQN layer $A$, it is first calculated as $A=(A+A^T)/2$. Next, $A_{Template}^{ASD}$, $A_{Template}^{NC}$ and $A_{contrast}$ are obtained by equations in Lines 423, 424 and 428 repectively. Finally, the top-20 edges with the largest weights from $A_{contrast}$ are selected for visualization.", "claims_and_evidence": "Appears to be.", "methods_and_evaluation_criteria": "The paper uses widely adopted datasets and metrics in brain network analysis for the empirical assessment.", "theoretical_claims": "Appears to be sound.", "experimental_designs_or_analyses": "1. More recent and relevant related studies should be included and compared for more convincing conclusions on the superiority of the proposed method. Some examples include \n[1] Cho, H., Sim, J., et al. Neurodegenerative brain network classification via adaptive diffusion with temporal regularization. In International Conference on Machine Learning (ICML), 2024.\n[2] Zhang, et al. A-GCL: Adversarial graph contrastive learning for fmri analysis to diagnose neurodevelopmental disorders. Medical Image Analysis, 90:102932, 2023.\n[3] Ma, Hao, Yongkang Xu, and Lixia Tian. \"RS-MAE: Region-State Masked Autoencoder for Neuropsychiatric Disorder Classifications Based on Resting-State fMRI.\" IEEE Transactions on Neural Networks and Learning Systems (2024).\n[4] Shehzad, Ahsan, et al. \"Multiscale Graph Transformer for Brain Disorder Diagnosis.\" IEEE Transactions on Consumer Electronics (2025).\n\n2. It appears that the performance on the two datasets are very different from those reported in the related literature, which raises the doubt of the validity of the results. For instance, the accuracy on ADNI/ABIDE reported on ALTER is around 67%/70% but that reported in the original paper is 74%/77%. Similar issues exist for ContrastPool with even larger gaps. What causes such a large difference in performance?\n\n3. The dataset ADNI used is small with around 100 samples. With such a small sample size, how would the proposed method adequately train a reliable model without severe overfitting? A convergence plot is expected to support this.  \n\n4. The authors should provide a detailed description of how they select the readout function for the final classification. Given their claim that BQN operates differently from GNNs and Graphormers, it is unclear whether traditional readout mechanisms remain valid for BQN. A more thorough discussion is needed to clarify whether the chosen readout is theoretically aligned with the proposed framework.\n\n5. For the case study, the authors should clearly explain how they transform the output of the final BQN layer into brain graphs. Specifically, it is unclear whether this conversion is based on a thresholding mechanism, top-k edge selection, or other criterion. Providing such details would enhance the interpretability and reproducibility of the method.", "relation_to_broader_scientific_literature": "This paper addresses an important open question: Is message passing truly necessary for certain graph-related tasks? Prior research has shown that in heterophilic graphs, message passing in GNNs can sometimes degrade node classification performance. The authors contribute to this discussion by providing both theoretical and experimental evidence in the context of static brain network analysis, suggesting that message passing may not be a crucial component for achieving strong performance in this domain.", "essential_references_not_discussed": "Many closely related recent works in brain network analysis are not included and compared (see the previous comment on \"Experimental Designs Or Analyses\").", "other_strengths_and_weaknesses": "Strengths:\n1. BQN demonstrates strong computational efficiency, making it a promising approach for large-scale brain network analysis.\n2. The design of BQN is relatively simple yet effective, striking a balance between model complexity and performance.\n\nWeaknesses:\n1. Limited baselines selected. \n2. Doubtful inconsistent empirical results.\n3. Small dataset used for ADNI. More large-scale datasets should be used.\n4. More clarifications on the readout function and the case study.", "other_comments_or_suggestions": "Nil", "questions_for_authors": "Please see the comments above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lvVkajB7Xn", "reviewer": "Reviewer_ufCW", "summary": "This paper investigates brain network modeling and identifies previous methods' shortcomings, including Graph Neural Network (GNN)-based methods and Graph Transformer (GT)-based methods. Specifically, they often use the Pearson correlation coefficients between pairs of ROIs (Regions of Interest) to construct brain networks, which function as node attributes and graph topology. Based on it, this paper introduces a novel Brain Quadratic Network (BQN) using the Hadamard product instead of the matrix product in previous methods. Extensive comparative experiments demonstrate the effectiveness and efficiency of the proposed BQN.\n\n## update after rebuttal\nThe authors have adequately addressed my concerns; therefore, I would like to maintain my positive evaluation.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1.Reproducibility is a concern. Although the authors claim significant performance improvements over state-of-the-art models with a relatively simple architecture, they have not provided code access, making independent verification challenging.\n\nR1. According to your suggestion, the source code has been made available at [https://anonymous.4open.science/r/BQN-demo](https://anonymous.4open.science/r/BQN-demo) for verification purposes.\n\n> Q2. Some of the related work mentioned appears to be extraneous or not directly relevant to the scope of this study. While Graph Neural Networks (GNNs) are foundational models for brain network analysis, it is unclear which specific models, especially SGC and APPNP in Eq. 3, are being utilized in this context.\n\nR2. We agree that the inclusion of certain related work, such as SGC and APPNP, may not be directly relevant to the scope of this study. While these models are foundational in the context of GNNs and their propagation mechanisms are indeed important for understanding message-passing models, their specific application in brain network analysis is limited. Therefore, we will adjust the manuscript to focus more directly on the models and methods that are most relevant to our study. Specifically, we will move the detailed discussion of SGC and APPNP to the appendix.\n\n> Q3.Although the meaning can be understood, the expressions \"with\" and \"without\" used in Figure 4 are not conventional. The authors are advised to adopt more standard notations, such as \"BQN\" and \"w/o residual\" for clarity. And in the caption of Figure 4, QBN is clearly a misspelling.\n\nR3. Thanks for your careful review. We will revise the legend in Figure 4 to \"BQN\" and \"w/o residual\" as suggested. Additionally, we will conduct a thorough review of the manuscript to correct any misspellings, including the error in the caption of Figure 4.\n\n> Q4. The statement of matrix multiplication is inconsistent. For example, the notation $\\mathbf{A} \\cdot \\mathbf{X}$ is used in Eq.3, while $\\mathbf{Z} \\cdot \\mathbf{W}$ appears in Eq. 4.\n\nR4. Thank you for pointing this out. We will revise the notation to maintain consistency. Specifically, we will update Eq. 4 to use the same matrix multiplication notation as Eq. 3, removing the symbol $\\cdot$. This will ensure uniformity in our mathematical expressions.", "claims_and_evidence": "The proposed BQN is well-founded and superior, as evidenced by both theoretical analysis and experimental results.", "methods_and_evaluation_criteria": "The proposed BQN makes sense for brain network analysis. It is novel in its utilization of a simple yet effective quadratic network.", "theoretical_claims": "After conducting a thorough review of the proof, I have essentially confirmed its correctness.", "experimental_designs_or_analyses": "The experiments are validated for their soundness. They are designed using widely recognized datasets and criteria. Performance is verified across diverse datasets, and hyperparameter analyses are carried out.", "relation_to_broader_scientific_literature": "Previous methods have primarily focused on designing specialized GNN and GT models for brain network analysis. This paper challenges this conventional approach, which is innovative and promising. It reveals that existing GNN- and GT-based models essentially rely on the generated Pearson correlation matrices in a redundant manner. To address this issue, this paper proposes an efficient Brain Quadratic Network (BQN).", "essential_references_not_discussed": "There is no other relevant literature that needs to be discussed.", "other_strengths_and_weaknesses": "**Strengths** \n\n1）The motivation of this paper is meaningful for brain network analysis. The authors are ambitious and reasonably challenge the rationality of message passing in previous methods. \n\n2）Figure 1 is presented with exceptional clarity and ease of comprehension. Figures 2 and 3 provide clear insights and design motivation. \n\n3）The proposed Brain Quadratic Network (BQN) is simple yet is underpinned by robust theoretical foundations.\n\n**Weaknesses** \n\n1）Reproducibility is a concern. Although the authors claim significant performance improvements over state-of-the-art models with a relatively simple architecture, they have not provided code access, making independent verification challenging.\n\n2）Some of the related work mentioned appears to be extraneous or not directly relevant to the scope of this study. While Graph Neural Networks (GNNs) are foundational models for brain network analysis, it is unclear which specific models, especially SGC and APPNP in Eq. 3, are being utilized in this context.\n\n3) Although the meaning can be understood, the expressions \"with\" and \"without\" used in Figure 4 are not conventional. The authors are advised to adopt more standard notations, such as \"BQN\" and \"w/o residual\" for clarity.", "other_comments_or_suggestions": "1）The statement of matrix multiplication is inconsistent. For example, the notation $\\mathbf{A}\\cdot \\mathbf{X}$ is used in Eq. 3, while $\\mathbf{Z}\\cdot \\mathbf{W}$ appears in Eq. 4. \n\n2）In the caption of Figure 4, QBN is clearly a misspelling.", "questions_for_authors": "Refer to Weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents a compelling and well-executed study that challenges the dominance of message-passing paradigms in brain network modeling, introducing the Brain Quadratic Network (BQN) as a simple yet powerful alternative. The authors provide both theoretical justification and strong empirical evidence to support their claim that message passing may not be necessary for this task, highlighting the effectiveness of Hadamard-based interactions. The work is thought-provoking, with the potential to shift prevailing assumptions in the field, and the proposed method demonstrates strong performance across benchmarks. Given the clarity of the analysis and the novelty of the insight, this paper is a strong accept and should be considered for a spotlight.
Decision: Accept (spotlight poster)