Beyond Pairwise Connections: Extracting High-Order Functional Brain Network Structures under Global Constraints
Authors: Ling Zhan
Junjie Huang
Xiaoyao Yu
Wenyu Chen
Tao Jia
Venue: NeurIPS 2025 poster ()
Keywords: functional brain networks
high-order interactions
graph structure learning
pairwise network modeling
Primary area: neuroscience_and_cognitive_science
TL;DR: We propose an end-to-end functional brain network structure learning framework to efficiently capture the high-order synchronizations of brain and strictly prove that this cannot be achieved by previous methods.

Abstract: Functional brain network (FBN) modeling often relies on local pairwise interactions, whose limitation in capturing high-order dependencies is theoretically analyzed in this paper. Meanwhile, the computational burden and heuristic nature of current hypergraph modeling approaches hinder end-to-end learning of FBN structures directly from data distributions. To address this, we propose to extract high-order FBN structures under global constraints, and implement this as a Global Constraints oriented Multi-resolution (GCM) FBN structure learning framework. It incorporates 4 types of global constraint (signal synchronization, subject identity, expected edge numbers, and data labels) to enable learning FBN structures for 4 distinct levels (sample/subject/group/project) of modeling resolution. Experimental results demonstrate that GCM achieves up to a 30.6% improvement in relative accuracy and a 96.3% reduction in computational time across 5 datasets and 2 task settings, compared to 9 baselines and 10 state-of-the-art methods. Extensive experiments validate the contributions of individual components and highlight the interpretability of GCM. This work offers a novel perspective on FBN structure learning and provides a foundation for interdisciplinary applications in cognitive neuroscience. Code is publicly available on https://github.com/lzhan94swu/GCM.

Beyond Pairwise Connections: Extracting High-Order Functional Brain Network Structures under Global Constraints
Ling Zhan Junjie Huang", Xiaoyao Yu', Wenyu Chen Tao Jia 1.24x
Coollege of Computer and Information Science, Southwest University, Chongqing, China Chonggiin Key Laboratory of Brain-Inspired Cognitive Computing and Educational Rehabilitation for Children with Special Needs, Chongqing Normal University, Chongqing, China
SSchool of Science, Guangxi University of Science and Technology, Guangxi, China
4 *College of Computer and Information Science, Chongqing Normal University, Chongqing, China 21 1e327eemail... swu. edu. en, sunjiehuang@swe edu. cn, xiaoyaoyu@email swu. en, cwy6108gxust. edu. en, tjia@swu. edu. cn
Abstract
Functional brain network (FBN) modeling often relies on local pairwise inter- actions, whose limitation in capturing high-order dependencies is theoretically analyzed in this paper. Meanwhile, the computational burden and heuristic na- ture of current hypergraph modeling approaches hinder end-to-end learning of FBN structures directly from data distributions. To address this, we propose to extract high-order FBN structures under global constraints, and implement this as a Global Constraints oriented Multi-resolution (GCM) FBN structure learning framework. It incorporates 4 types of global constraint (signal synchronization, subject identity, expected edge numbers, and data labels) to enable learning FBN structures for 4 distinct levels Ssanple/suhjec//rruuppprreee olution. Experimental results demonstrate that GCM achieves up to 30.6% improvement in relative accuracy and 96.3% reduction in computational time across 5 datasets and 2 task settings, compared to 9 baselines and 10 state-of- the-art methods. Extensive experiments validate the contributions of individual components and highlight the interpretability of GCM. This work offers a novel perspective on FBN structure learning and provides a foundation for interdis- ciplinary applications in cognitive neuroscience. Code is publicly available on https://githuu. httgggttuub.... wtt.a//cco.///aanss////aC
of modeling res-
Introduction
Functional brain networks (FBNs), which model inter-regional coordination graphs, have become
a central approach for understanding cognition, behavior, and mental disorders [
2 They are
typically inferred from multivariate time series (MTS) recorded by fMRI EEG [ 4 with the
goal of capturing the inter-regional dependencies embedded in neural dynamics 6. However, because these dependencies are only indirectly observable, reliable network construction remains challenging [8] [9.
Traditional methods construct FBNs by estimating pairwise statistical interactions (e.g., correlation, coherence) between regional signals [0]
13
Recent machine learning models extract
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Pairwise vs High-Order Interaction
Local vs Global Constraints
Adjacency
Brain Region Activity
Pairwise Interaction
Region
Active
inacirvc
Region
Region E
AcTiC
corriA, = corr/B, C)
Local Constraints
Inactive
Activ
Node Features
Active
Active
corriA =
Inactiin-
Tratther
Inaatiinn
Signal Syndhronzaatin
Subject Idcerity
Adjacency
Quantification
High-Order Interaction
Region A
Region B
[1. 0, 1, 0]
[0, 1.
Global Constraints
Edge Numbers
Expected
Update
Region
[1, 0, 0)
Data
Labels
Previous: Pairwise Model
Ours: Extract FBN under Global Constraints
Multivariate Time Series
Adjacency
Multivariate Time Series
FBN Prototype
Adjacency
Local Constraimts
VS
GNN
Holistic
Feature
Adjacency
Encoder
Gradient
Local
Local
Global
Constraints
Constraints
Constraints
An illustration of the two paradigms for FBN modeling. Top: Conceptual diagrams illustrating the core theoretical differences. The top-left provides a concrete example of high-order (XOR) interaction demonstrate how pairwise models fail to capture patterns that high-order models can identify. The top-right contrasts the scope of Local Constraints (acting on single nodes or edges) versus Global Constraints (acting on the entire network). Bottom: Comparison between the previous pairwise paradigm with our proposed GCM framework, which treats the adjacency matrix as a single, leamablee object holistically shaped by global objectives.
Figure I:
latent features from MTS via neural encoders, but still rely on pairwise computations to form adjacency matrices for downstream graph analysis [4.. Despite current shift toward end-to-end modeling 8 15 16), fully connected graphs could reinforce their performance in prediction according to our empirical study (Appendix F). This suggests that the initial structures generated by pairwise methods may be suboptimal or even detrimental. As illustrated by the XOR example in Figure (Top-left), a brain region might activate only when its two driving regions, A and B, are opposite states (one active, one inactive)-a relationship that cannot be described by any pairwise correlation between (A,C) or (B,C) alone. This type irreducible, multi-way dependency is typical example of high-order interaction. Recent evidence from hypergraph modeling also suggests that it is necessary to model high-order interactions in the brain [17 [8] [9] which cannot be fully captured by pairwise interactions proved in Section 4 However their inferred FBNs agnostic to analysis tasks and thus need further statistical processing. Meanwhile, these methods are practically limited by their strong assumptions prior knowledge demands on dynamics 20 22 and high computational
complexity 23 [24].
Motivated by the observation that high-level cognitive demands shape the organization of functional brain networks 225 26). we propose to extract FBNs from MTS under global constraints, as illustrated in Figure [ (Top-right) In this paradigm, we distinguish between two types of constraints. Local constraints operate on a mode-pair basis, such as using Pearson correlation to determine a single edge weight, where the final graph is aggregation of many independent, local decisions. In contrast, global constraints are top-down priors that shape the entire network structure holistically, guiding the optimization of the adjacency matrix single entity rather than a collection of independent edges. In this way, FBNs can be learned efficiently and constrained by input MTS and global priors directly, without relying predefined dynamics [9].
Therefore, we design the Global Constraints oriented Multi-resolution (GCM) framework as an implementation of this paradigm. As illustrated in Figure (Botoom, comparing to previous pairwise models, in GCM, FBNs are treated as structured entities optimized under the holistic influence of our global constraints, derived from both data and task semantics, rather than byproducts of local

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interactions. To ensure interpretability, GCM requires clear semantic specifications on the learned structures and global constraints. As for structure, it essential to define the modeling resolution, Depending on the analytic goal, GCM supports four distinct resolutions: sample (e.g., short-term activity (29)., subject (e.g., individual traits [30]. group (e.g., cohort-level patterns [31] D, and project (entire dataset, e.g..
i.e., the unit of representation on which the graph structure operates [27 28
population level abstractions (32)). ). To the best of our knowledge, the semantic distinction between these resolutions has not been well-addressed. However, without such explicit resolution specification the learned FBNs risk semantic ambiguity or misalignment with downstream tasks [3 Our theoretical and empirical analysis validate the necessity of this formulation.
As a support to the concept of global constraints, recent work in control theory and cognitive science has proposed a perceptual control architecture [35.. In this framework, organismal functions adapt to environmental disturbances by integrating diverse sensory feedback into coordinated internal responses. Inspired by this view, GCM models the brain's 's functional structure as an adaptive variable shaped by multiple global constraints as signals. These constraints, which we define as global because they influence the entire network structure simultaneously, reflect distinct form of global feedback, while their joint influence is integrated through a unified gradient. Specifically, our framework integrates four types of global constraints. (1) Signal Synchronization: The model is end-to-end, meaning the learned graph is inherently constrained by the high-order temporal dynamics within the raw signals. (2) Subject Identity: This constraint imposes consistent intra-subject signature, or "neural fingerprint," through contrastive regularization [6). (3) Expected Edge Numbers: An adaptive constraint ensures global sparsity in the learned network (37. (4) Data Labels: Task labels guide the structure to align across samples via a supervised loss. These constraint signals integrated through gradient and fed back through propaaro,,,,,,nn with Gumbell-Sigmoin (38) enabling differentiable sampling over binary graphs. Meanwhile, to retain local co-activation patterns, GCM integrates graph neural network (GNN) as its backbone. Our empirical results report the contribution of each module.
Our contributions are threefold. First, we provide a theoretical proof that pairwise-based models are mathematically incapable of recovering the high-order interactions present in MTS, thereby establishing the necessity of a new paradigm, such as our proposal of learning FBNs under global constraints. Second, we implement this paradigm as GCM that directly learns discrete FBNs under global constraints specific to four semantically separated modeling resolutions. Third, extensive empirical evaluation confirms that GCM not only improves classification performance but also enhances the interpretability and semantic alignment of the learned FBNs.
2 Related Work
Because the idea of high-order FBN structure is addressed in hyper network modeling, and the design of GCM is inspired by the ideas of Brain Graph Interpretation (BGI) (39] and Graph Structure Learning (GSL) [40, we briefly introduce the related works along these aspects in this section.
Hyper Network Modeling Research on FBNs aims to model and understand the cooperative functions of the brain and has prospered with the development of complex network theory 6 Pairwise-based network modeling is a relatively mature field and has been approached from numerous
perspectives utilizing diverse tools both in network science and machine learning 9 4T
42
43
44
Comparatively, hypergraph inference is still in its carly stages, but the field is evolving rapidly [2
45 46.
Recent efforts span probabilistic modeling grounded in historical connectivity 20 47 or
contagion traces [48], as well as optimization-ban these advances, their computational overhead presents major bottleneck for scaling to real-world datasets 24). More recently, a pioneering work proposes an end-to-ent end model for learning a fixed number of hyperedges from individual samples [4 In contrast, our proposed GCMframework is explicitly designed to learn a single, generalizable FBN that represents an entire cohort (e.g., at the subject or group resolution), distinct goal focused on cross-sample interpretability.
formulations adapted to MTS 24]. Despite
Brain Graph Interpretation Brain Graph Interpretation (BGI) methods aim to identify and select
important edges in FBNs that contribute most to predictive outcomes [50). These approaches
typically construct FBNs based on pairwise interactions and learn gradient-updated masks to filter
structures. Some methods refine the masks dynamically during training [51 52), others generate
3

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predictive subgraphs to facilitate interpretation [39 53 54 integrate multiple modalities for cross- validation [2
55 [6]. or employ attention mechanisms as trainable criteria for edge selection [57] These methods rely on pairwise-constructed FBNs and apply post-hoc explanations without structural optimization. Consequently, their explanations are often resolution-agnostic lacking explicit differentiation across semantic levels. In contrast, our framework directly models functional structures under global constraints at multiple modeling resolutions.
58
59
Graph Structure Learning Graph Structure Learning (GSL) treats imput network structures as noisy or incomplete and aims to refine them based on node features and initial graphs [40]. Traditional GSL research focuses on defending against adversarial attacks [60] [[], later evolving toward task-
specific structure optimization 662 mostly for node-level tasks on single graphs [6
64
65|
66
67 [8]] Recent benchmarks [69 highlight that only a few models, such as VIB-GSI 770 and
HGP-SL 71], address structure learning across multiple graphs. In brain network modeling, Zong
et al. B introduced GSL to FBN learning by aligning brain regions using a diffusion model and constructing structures based on feature correlations.
Unlike prior methods, GCM supports end-to-end extraction of high-order FBN structures by explicitly encoding modeling resolutions and global constraints, aspects often neglected in existing approaches.
3
Preliminaries
We begin by defining a "sample" as the mini- mal unit of MTS (e.g., single EEG or fMRI fragment). Formally, each sample is a matrix X E X' with shape N x T, where N is the number of nodes and T is the number of time stamps. Each FBN structure is an undirected graph G = (V,E) with [V| N, where V de- notes node set and & denotes edge set.
Sample-level
Subject-level
Group-level
Project-level
1 Sample
Multiple Samples Subject
Each Group Multiple Subjects
All Data
Levels of modeling resolution. We consider four levels of modeling resolution, which spec- ify how many samples are aggregated when forming single FBN:
Modeling
Resolution
Sample Amount
Figure 2: Illustration of the four modeling resolu- tions defined in this work. Each resolution corre- sponds to distinct level of structural abstraction and data aggregation.
Sample level: Each sample corresponds to one FBN, representing the network within a single recording (e.g. dynamic FBN) (29). Subject level: All samples from the same subject share one FBN, reffectingggggeee.......roee relatively stable brain state
(c.g. brain fingerprinting) 330 Group level:
Samples from group (e.g., clinical cohort) form one FBN, capturing representative pattern of that group (e.g. cognitive states) B1. Project level: The entire dataset yields single FBN, often
used as statistical template for the project (e.g. brain atlas) (32). Formally, research level defines a mapping function X + G, which aggregates one or more samples into an FBN. Appendix B rigorously proves that sample, subject, group, and project FBNs are semantically non-equivalent.
FBN Structure Learning. We model FBN structure learning problem as extracting connectivity structures from MTS specific to the modeling resolution. Formally, let P(x)) denote the distribution of samples and P()) the corresponding label distribution. Our goal is learn a parameterized family of graph distributions P(g o. (()),, where ((Ø) are learnable parameters conditioned on the chosen research level mapping ø. Each distribution is induced by the adjacency matrix distribution ((A o. (())), where adjacency matrices A E IR N encode expected connectivity probabilities (p((ppee Φ, (())). Here Ey.., denotes an edge between nodes u and v in the FBN, and p(eu,e Φi ( ()) flexibly models connectivity without assuming specific parametric forms. Thus, by learning PCG Φi, (())) under different modeling resolutions d we enable a multi-resolluiion analysis of functional brain networks.

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The Limitation of Pairwise Network Model
In this section, we formally prove that pairwise network model cannot fully catch the high-order coherence encoded in MTS. Our analysis aligns with and extends prior findings on linear consensus dynamics 72 and triadic interactions [2]]
4.1 Definitions
Let (t)
Xx())) be N --diimeniional, zero-mean, finite-variance MTS. For each lag w & denote the second-order moment Ei((w) structure captured by the k-th order cumulan
[w]] Higher-order joint
Ki (WI, ( Wk-1) cum{ Xi, (1), Xig (1 + wi), .... Xi (1 wx-]]
Proof. pairwise network an adjacency matrix E RNN whose entries are generated by Ag = fiy( Xx), where fij is specific edge rule depending only on the bivariate distribution of Xx,,
Higher-order dependence. We say that X exhibits higher-order dependence if there exists & 2 3 and indices i,....,44 such that Be, ii (.) # O.
4.2 The Expressive Limit of Pairwise Network Models
Lemma (Second-oriic sufficiency). A random vector Z is multivariate Gaussian iff all cumulants of order k 3 vanish (73.. Consequently, only for Gaussian data is the joint distribution fully determined by second-order statistics.
Theorem (The limitation of Pairwise Network Model), Let X and ( be two d-dimensional time series (d 2 3) satisfying 2ij =(jW)
D (w), Vi, to E for some finite lag set 'T actually used by the edge rule. Assume further that there exists ls 2 3 with
(1) ()) * ()) for some in,...,
(2)
Then every pairwise network generator Au- XX, produces A A2) while the two joint processes (1) law (2), (Proof in Appendix C
Corollary 1. Theorem remains valid for weighted graphs (let be identity), directed graphs, or any edge rule that aggregates over a finite lag set Q.
Methodology
We introduce the proposed GCM in this section. It consists of three modules: (i) prototype- graph generator with humbel-Sigmoid relaxation, (ii) a batch-wise binarization algorithm (BBA) that enforces hard edge sparsity, and (iii) multi-objective training loss aligned with task labels, subject identity, and sparsity priors. Appendix summarizes pseudocode for GCM and BBA.
Prototype Graph and Continuous Relaxation. We learn binarized functional brain network (FBN) AE E (0, 1) NxN from the conditional distribution P(A o. (())), where ó parameterises the graph sampler and (()) denotes GNN weights. Instead of drawing A directly, GCM first samples prototype matrix Á ~ P(Å d. (())) and symmetrises it by A of À + ÅT). where a(.) is the
element-wise sigmoid so that Aij {0, 1]. We initialise A.j iid. ((.
To enable back-propagation through discrete edges, we adopt the relaxation 38). Let M,, log(-- log u). u~2((0, 1); the relaxed adjacency is Å (A)M)/7)
(Concrete)
where 7 is the temperature. Gradients w.r.t. and 0 are obtained via the chain rule.
2 Any other faithful measure of k-variable dependence (e.g. joint mutual information, total correlation) would equally well.
5

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Batch Binarization Algorithm (BBA). Given mini-batch A R Rx NxN BBA keeps the k(O)
largest now-major flatted) entries in each A (b) where k(()) 2/2 - Eه A is the expected
edge number of (0) updated during the training process to ensure an adaptive constraint of sparsity, and is the index of network within the batch:
(b) m (b) Q())
(b) = 1 1(jETop-k(0)
followed by reshaping back to obtain a binarized batch pp
Graph Representation Learning. For each sample (X, 9), node signals X + RNxx are fed to L-layer GNN backbone (74]. After message passing, a permutation coamutation-invariann invariant readout p(-) (mean, sum, max, or Transformer pooling (75)) produces graph embedding e E
Training Objectives. Label-aligned loss: A linear classifier predicts class logits ý, softmax W TyTe, e; + b), optimised by cross-entropy Li - Di log Wis Subject identity contrast: Samples from the same subject are positive, others negative:
C2
Σ
logor
T Tal
1
log of T.
Ppooo
Paegl
(i.j)EP pos
(f.t)
with temperature Tel. Sparse prior: We impose an & surrogate on Å as R
|||||,
The total
objective is C = Li + aL2 + 3R, optimised jointly over (0, A) using Adam.
Theorem 2 (Existence of a Higher-Order-Fxact GCM). Lei (() E RN be stochastic process and y E (I....,C) obey
y = g (f), (I), k 2
(5)
where (i) vb I2 + R is measurable: (ii) g- is class-separable measurable map (assuming 3 disjoint measurable pre-images for each class). For any & >> O there exist &f 8 temperature T >> 0, and depth L. max(1, diam(A") A)) such that the GCM classifier fo satisfies
Pr o (X(1) + v S.
This theory ensures the existence and convergence of GCM. Our formal proof of Theorem 2 in Appendix D shows that GCM can recover arbitrary high-order rules. These justify both our multi-resolution design and the use of global optimization.
Experiments
To demonstrate the advantage of GCM, utilize it with other baselines in each classification task. The FBN structure learned by GCM is used as the computing graph for a GNN classifier. GCM outperforms other baselines using the same GNN layer, the difference can only be explained by the underlying network structure, hence proving that GCM learns a more accurate and comprehensive representation of FBN. For the task the sample level, we also consider several none- GNN based methods for their prevalence in neuroscience. The performance advances of GCM over them validate the benefits of combining GNN with extracting high-order network information.
Table 1: Statistics of the Data Scts
#samplc
#subjcct
#class
#node
#tcature
Class-Type
DynlICP Autivty
260.505
7,443
sample
DynlCCPs, ago
36,278
1,067
100
100
subject
DynHCPaaser
36,720
1,080
subject
Cog State
90,000
60
5
61
300
sample
SL.M
3,246
541
236
30
subject
6.1 Experiment Settings
Data We use 5 open-source real-world datasets, including DynHCP Activity/Age/Gender [76), Cog State [77), and SLIM (78). These datasets cover both fMRI and EEG, the most commonly used brain imaging modalities, for testing broad applicability. The overall statisties of the data are summarized in Table I Description and preprocessing of the data are detailed in Appendix!]

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Baselines We choose traditional methods (LR, SVM, Random Forest (RF), XGBoost, MLP). which are commonly used in neuroscience domain, establish baseline performance. GNN families (GCN (74), SAGE 79. GAT [0) GIN (I) are included to highlight the improvement of GCM against GNN backbones. To position our work within the current frontier of structured graph learning, we include representative SOTA methods from both the BGI line (IBGNN (82). IBGNN+ [82], IGS [5]], IC-GNN [53], BrainIB (39) and the GSL line (VIB-GSL 70), HGP-SL 771, DGCL (8). To further benchmark our method against the SOTAs, we incorporate two key baselines. First, we include the Brain Network Transformer (BNT) S3] due to its remarkable predictive performance.
Second, we include Hybrid [49], conceptually related model that learns fixed quantity of high-
order interactions. (Details in Appendix K
Tasks Brain activity prediction tasks are commonly categorized into inter-subject and iutra-subject classification 88 [85 In inter-subject classification, all subjects and their samples are split into training, validation, and test sets, with predictions targeting the test set subjects' labels. In intra- subject classification, each subject's samples are split into training, validation, and test sets, with predictions focused on the test set labels. We align the baselines with the proposed four modeling resolution levels for fair comparison, as defined in Preliminaries, based on their design. For all levels, the task is to predict the label of each individual sample. For GNN-based methods, the learned FBNs used as computation graphs, where each sample serves the feature of node. Appendix Contains more detailed reproducibility settings such as hyper parameters and experiment environment.
6.2 Experimental Results
We report the most important results to support the superior performance of the proposed GCM and the interpretability of the learned FBN in the main body. We also provide more detailed analyses including The Role of Initial prruuture(Appendixxe (Appendix G.. FBN Structure Visualization Appendix II and Sensitivity for a more comprehensive assessment of GCM.
Results Using Additional Metrics
H
Table 2: Sample-Leve Structure Learning Results (Accuracy+su)
Mothod
Pynty
DysHCP
DysHCP Gaile
CogState
SLIM
Intra
Inter
Intra
Inter
Intra
Inter
Inim
Inter
Inmm
Inter
LR
0.80M
0.8792000s a.357
0.625 21
0.218 0.276
2.
RF
2.94520.022
2.825
0.112000
0 0.00
0.824
0.629 25.
0.268
0.264
0.6352000
0.62620.am
XCiRoost
D.959 2.05
0.832.0001
0.74520000
0.388
0.798
0661
0.307
2.30770000
0.2822a.03
06
0.610.0.00
SVM
0.827 a..42
0.39
0,882
0631acaa7
0.321 anoss
0.270
0
0,567
0.554
MLP
0.
0.788gomm
0.372
0.903
0.278
0.518
0.5492000
GCN
0.774
a.736
1.425
0.45 abucso
0.650
0.54
0.303
0.552amm2
0.543
SAGE
go
20434
0.633
0.544.0
up ase
0.35
0.301 03.000
066720m2
GAT
0.138
0.202
a0/c04
0.542
0,532
0,305
L0.00
1.622
0.602
GIN
0.480
0,495
0:449
0.401
0,572ammr
0.565 1885st.m
0,280 an3
0 0.5
0.
03465
VIB-GSI.
D.925
2.600
0.429 40c31
0.781
0.574ac.034
0.303 a0
4.529
0.537
HGP-SL
0.647 an.00s
0.529
0 0.444
0.766
0612ac..0
1.286
0.270
0 0.556anm
0,552.0.0
DGCL
0.758
g.392
0.860
0615
0.316
0.552an 0.275
0
IBGNN
2..08
0.768 2.28
a.39
0.858
1.343
0.300
0.68
2.617.0027
IC-GIN
D.678 0.
2.85
0.68
0.318
0313
0.6840 no
0.614.0m
BrainIB
0.826
0.899
0,818
0.44
g.045 apcco
0.873
0.662
c..00
0.352
0.325
.0.22
0.6
0.631 1o
RNT
0.974
2
0.945 2000
0
0.386
0,678
0.337
0
Hybrid
0.801 20.13
0.76610.000
a0.05
0.442
0.788
0.592
00002uc.00
0.203
0.286
0.53740.00
0.522
GCM,
0.963g0.00
0.82
0.855
0.463
0.937
0.668
0.352
0,356
0.600
0.658.o.am
Musolt-Reoolutinn Structure Learning: Sample Level The sample level is the most common scenario in brain activity prediction, as it focuses on individual samples and their corresponding brain network structures. At the sample level, GCM consistently outperforms or matches traditional methods and SOTAS as shown in Table 2 These results highlight that GCM excels in capturing the underlying brain activity structure, surpassing Pearson-based FBNs, which are limited by their reliance on linear correlations. On the one hand, the results prove that catching higher-order synchronization of MTS is beneficial to downstream prediction. On the other hand, this demonstrates the effectiveness of GCM in learning a more robust and flexible structure that better represents the data's complexity. GSL and BGI SOTAS gain an overall superiority against GNN-based methods, providing the necessity of FBN structure learning. An interesting phenomenon is that traditional methods exhibit strong competitiveness on several datasets. This explains why these methods are still popular in current neuroscience domain.
7

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Table 3: Multi- musti-Resolution Structure Learning Results (Aceuracy =nd))
Method
DyaHCP
DyaHCP kg
DynlCPowwer
CogStre
SLIM
Intra
Inier
Intra
Inter
Inira
Inter
Intra
Inter
Inira
Inter
0.861 anccee
0.808
0.328
0 2ooon
GaMaatee
0.889
20.000
0.478
0,939 20002
0.364
10.00
0.674
IBGNN+
0.763 10
7888
0.881 20oss
2.356
GCMprjee
0.969
0.869 a0.0s
0.878 20.002
0.878
0 3..62
0,933 sou007
0.674 0.674
0.437 ano
0.419 20.023
0.644
GCM
0.971
0.855 201
0.891 090
0 0,489
0,958go
0 686 c56
0,403 oos
0,454 013
0 0,00 ocs
0.694
Multi-Resolution Structure Learning: Subject, Group, and Project Levels Comparing to sample level analysis, methods in these three levels are less discussed in previous FBN structure learning works. Thus only two competitors respectively aiming for subject-level and project-level are included. As shown in T GCM outperforms these two methods on most conditions. This not only shows that GCM adapts seamlessly to higher-level modeling resolutions, but also proves that encoding and utilizing global constraints is crucial for tasks in these level of resolutions. It should be noted that we isolate the results of group-level GCM. On the one hand, there is no previous discussion particularly aiming at this scenario. On the other hand, FBN learned in this scenario may suffer the argument label leakage under the context of machine learning settings. However, the target in this scenario is to Iearn possible FBN structures that can better reflect the structural distinction between categories. Thus, the accuracy is not only a judgement for method, but an index to investigate the confidence that the learned FBNs reflect the non-linear relationships between the input data and labels. From this point of view, although GCM goop literally outperforms all the baselines, there's 's a large room for improvement to learn reliable FBNs in this scenario.
Time Complexity across Different Methods and Data sets
153
14V775
Tihee
307
10.1
30'
5:54
IBGNN
EGNN+
IGS
C- GWN
Broin
HGP SL
V/B GSL
DGCL
GCM
Figure 3: Time consumption of each method.
Computational Analysis GCM's computational efficiency is primarily determined by the GNN component (here, SAGE), incurring O(N2) complexity. Figure 3 compares computation times of BGI and GSL methods, including GCM, across datasets. All GCM variants share identical costs, so they appear as one category. We omit DynHCP Age and DynHCPGemter since they match DynHCP Activity in graph dimensions and SLIM in class counts. The superior efficiency of GCM is clearly reflected in the results, owing to its streamlined design.
6.3 Ablation Studies
Impact of Batch Binarization Algorithm (BBA) Table presents an ablation study of our BBA to validate its key design choices. First, the binarized GCM model consistently outperforms non-binarized "Dense" variant across most tasks and resolutions. This confirms that binarization acts as a crucial regularizer, forcing the model to learn sparse structural scaffold and preventing overfitting. Second, GCM with its adaptive BBA also surpasses all the performance of fixed thresholds plateaus as density increases, BBA avoids this issue by adaptively tuning sparsity for each FBN; nonetheless, a 5% fixed density offers a reasonable trade-off for simpler applications. These advantages are consistently observed across all resolutions (sample, subject, group, and project), validating the BBA module as a critical and robust component for learning high-performing FBN structures.
methods, While
8

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Table Ablation Study of Structural Modeling Choices (ACC)
Sample-level Ablations
Subject-level
Group-level
Project-level
Dataset
Task
Dense
0.05%
0.1%
0.5%
1%
5%
GCM
Dense
GCM
Dense
GCM
Dense
GCM
DyaHCPAsssty
Inter
0.843
0.792
0.808
0.827
0.835
0.837
0.852
0.849
0.857
0.886
0.855
0.857
0,869
Intra
0.950
0.895
0.927
0.943
0.949
0.951
0.963
0.962
0.963
0.971
0.971
0.959
0.969
DynHCPAs Age
Inter
0.408
0.381
0:402
0.418
0.420
0,428
0,463
0.425
0,473
0.432
0.489
0.418
0.462
0.837
0.845
DynHCPosste
Inter 0.657
0.605
0.628
0.631
0.633
0.653
0.668
0.662
0.670
0.673
0.686
0.637
0.674
Iotra
0,889
0.883
0.912
0.928
0.937
0.939
Cog State
Inter
0.359
0,311
0.313
0.321
0.325
0.327
0.356
0.357
0.364
0.345
0.493
0.350
0.419
Intra
0.356
0.322
0.333
0.343
0.348
0.348
0.352
0.392
0.413
0.482
0.493
0.399
0.437
SLIM
Inter Intra
0.656
0.604
0.622
0.631
0.632
0.639
0.658
0.670
0.674
0.683
0.694
0.657
0.644
0.661
0.618
0.632
0.637
0.645
0.649
0.680
0.665
0.681
0.677
0.692
0.670
0.655
Impact of Subject Consistency Contrast The impact of subject identity is reflected by training strategies GCM. Results are shown in Figure + Specifically, GCMi represents GCM without individual consistency loss L2, and GCMd refers to GCM without sparsity regularization R. Re- moving L2 and R results moderate performance drop under most conditions. However, for some settings, such as in group level and project level prediction in inter-subject prediction, the degradation becomes unbearable. This emphasizes the necessity of maintaining subject consistency and sparsity in the FBN structure.
6.4
Case Study: Sex Difference
(a)
Sample Level
Subject Level
Group Level
Projest Level
1.9
(b)
8.--
Somday
Sattendae
Cip Plet Ray
Ashob
Google
nto
Aatstty
Cop Shot
Figure 4: Accuracy of GCM using different training strategies.
Figure 5: FBN of DynHCP combin- ing four modeling resolutions, with brain regions as chunks and edges as chords.
To illustrate the necessity and interpretability of our proposed four-level brain network division, we visualize FBNs learned by GCM on DynHCP in the intra-subject setting, where model confidence (accuracy) is highest. For sample and subject levels, we average the networks of male and female subjects to obtain group-wise representations. Following the original release (76), we retain the top 10% of edges by weight and visualize combined networks at the sample, subject, and group levels. ROIs are grouped into seven brain regions based the Schaefer atlas (86).
As shown in Figure 5 the common edges between male and female specific each resolution are depicted in gray. The edges particularly shown in female and male across resolutions are depicted in red and blue, respectively. Common edges between female and male across resolutions are highlighted in cyan. The results align with previous studies in the following three aspects: (1) Higher-orde order synergies exhibit non-random organization, integrating multiple brain regions into coherent systems that appear independent under conventional correlation-based analyses [17. (2) Under each modeling
resolution, most of the edges are shared among female and male 87 (3) Male FBN shows higher
edge weights in edges linking to vision networks [88) while female FBN shows more complex

---

pattern 89 The limited amount of common edges sharing across different levels further shows the uninterchangeabiiity between modeling resolutions as illustrated in Theorem 3 which the first time to be clarified to the best of our knowledge. Comparisons between FBNs learned by GCM and baselines are provided in Appendix
Convergence Analysis
To demonstrate the convergence of GCM on intra- and inter-subject classification tasks, as proved in Theorem 2, present the training loss curves for cross-entropy loss (C₂) and contrastive loss (C2) in Figure 6 Both losses eventually converge on each dataset. It is interesting that L1 shows temporary vibrations, aligning with the structural reconfiguration and path reselection process observed in FBN evolution (35 L2 exhibits a slower, linear decline, reflecting a cold start in subject-leve learning. These results highlight opportunities future enhance the utilization of global constraint.
DynHCP Activity
DynHCP Age
DypHCP Gender SyndCr
state
sim
2.0
3.-
1.22
0.30
2.16
Inter
1.65
inter
-882
32
10
0.0
Trai Concraation Losa
Tean Casa Lasa
Figure 6: Training loss of GCM for each dataset reaches steady state after training several batches.
8 Discussion and Conclusion
Our results support the theoretical expectation that global constraints enhance structural guidance, surpassing pairwise-based methods aligning information flow across raw data, macro attributes, and the learned FBNs. The learned high-order patterns align with previous conclusions in hyper graph modeling, proves the interpretability of the learned FBNs. Notably, we observe that functional brain networks learned at different resolutions exhibit distinct structural patterns, validating that these modeling resolutions are not interchangeable. While the proposed GCM framework offers strong empirical and theoretical benefits, it also has limitations and potential societal impacts as summarized in Appendix L and Appendix M respectively.
Instead of treating prediction accuracy as a unified metric for model performance 82 we reinterpret accuracy as confidence index that reflects the reliability of the learned FBNs. This formulation offers principled way to compare models across different analytical objectives and renders overfitted FBNs at the group level as deliberate and meaningful modeling outcomes rather than incorrect label leakage. This further extends traditional statistical comparisons of individual- and population-level connectivity into a unified structural learning framework [90 91].
Moreover, our study offers a thcoretically grounded implementation of the proposed global updating modeling, which provides reliable framework for FBN structure learning. By analyzing the limitations of existing approaches, we provide a theoretical complement to MTS-based network modeling field [3,92] Moreover, by exploiting how evaluation metries are utilized, and by rethinking overfitting phenomena in group-level modeling as a reliable modeling results, our work introduces a novel lens for applying machine learning in neuroscience and cognitive science studies. Future work includes addressing the limiations above, or extending GCM to multi-modal and multi-view scenarios [93]. Further exploration on this topic not only facilitates brain science but also sheds lights on interdisciplinary research.
10

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Acknowledgements
This work is supported by Natural Science Foundation of China (No. 62402398, No. 72374173), the Fundamental Research Funds for the Central Universities (No. SWU-XDJH202303, No. SWU- KR24025), University Innovation Research Group of Chongqing (No. CXQT21005), China Scholar- ship Council (CSC) program (No. 202306990091, No. 202406990056) and Chongqing Graduate Research Innovation Project (No. CYB22129 129, No. CYB240088). The experiments are supported by the High Performance Computing clusters at Southwest University.
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A
Notation Summary
Table Al: Summary of notations used throughout the paper.
Symbol
Description
General & Graph Theory
N
Number nodes (brain regions) in graph. Number of time stamps in single MTS sample. An undirected graph representing an FBN structure. The set of nodes and edges in graph, respectively. An edge between nodes u and v.
T
GV(VE
V.E
Cu,v
Input Data & Modeling Resolutions
X E NNT
A multivariate time series (MTS) sample. An N N-dimensional MTS time t.
X(L)
60)
A mapping function that defines the modeling resolution.
P()).P())
The distributions of samples and their corresponding labels.
GCM Framework Components
A E (0,1)
The binary adjacency matrix of learned FBN.
Ä
The learnable, continuous prototype matrix for the graph generator. A matrix of Gumbel noise used for Gumbel-Softmax relaxation.
M
A
The relaxed (continuous) adjacency matrix after Gumbel- -ootbel-Softaa.. The temperature hyperparameter for Gumbel-Softmax. The batch size.
B
kb(
The expected number of edges for the b-th graph in a batch.
L
The number of layers in the GNN backbone.
p()
A permutation-invariant readout function (e.g.. mean, sum).
E TR..
The graph embedding vector produced by the readout function.
Training Objectives
(())
The learnable parameters of the GNN, conditioned resolution ø.
y:
The predicted class logits for sample The label-aligned cross-entropy loss.
C1
L2
The subject identity contrastive loss.
Tel
The temperature hyperparameter for the contrastive loss.
R
The & sparsity regularization term on A.
8
Weighting coefficients for the contrastive loss and sparsity term.
The total objective function for GCM.
Theoretical Analysis
Eij(w)
The second-order moment (covariance) between signals X.(() and X₃ w).
The k-th order cumulant, used to capture high-order dependence. An edge rule for pairwise network, depending only on two variables.
(..)
X (2)
Two time series used to prove the limitation of pairwise models. A measurable function representing a high-order interaction rule. A class-separable mapping function in the proof of Theorem 2 In proofs, the set nodes on which a ground-truth rule depends. In proofs, the oracle graph structure (e.g., k-clique). Enhanced feature for node v, combining signal and an ID embedding e The set of node features for all nodes in the set S, i..., {2,|v E S}.
v())
(.)
A*
2, = (t)
Es
B
Semantical Uninterchangeablltty of the Proposed Scenarios
We cast the four resolutions of functional brain network (FBN) learning (sample, subject, group, and project) into formal hierarchical model and prove the four random FBN variables are mutually non-equivalent in distribution whenever inter-level variance is non-zero.
24

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Hierarchical generative model. Let G(P) Pg be the project-level FBN, G Poljl g(vl) the m-th group FBN, Gli Ppjol- Gil)) the n-th subject FBN in group m. and g Polol gmin) the r-th sample (d for "dynamic") network. We assume
Var [ar[
[(P)] o > O, Var Var [ath
G = o > O, Var (ggh
mi) = o 0. (1)
Lemma 2 (Non-degenerate variance implies distributional gap). If Var(a | > 0 almost surely, then X and y are not equal in distribution.
Proof. Equal distribution would imply Pr|x [ = which forces conditional variance to zero contradiction.
Theorem (Mutual non -on-eeuivalence). Under Eq. B the four random graphs gld), g gla), and g(r) are pairwise non-equiin non-equivalent in distribution; hence they encode genuinely different semantics.
Proof. Apply Lemma 2 transitivity yields all six pairwise inequalities.
successively: 03 0= g( $ g()) of -0=0 0=0g g(()) g(o), and so on;
C Proof of Theorem 1
Proof. By Lemma] T a random vector is fully determined by its second-order statistics and only if it multivariate Gaussian; for non-Gaussian vectors,
match all second-order cumulants
while altering higher-order (S 3) cumulants. Let ( and X (2) satisfy
x (w) = x (w), Vi,J, VWED,
(1) but assume there exists some order A: 2 3 such that Ki () * ( ())
Since each pairwise network generator Aij lay (X;, X,) depends only on second-order information
derivable from the bivariate distribution of (X,, X,) over ft matching all second-order moments
implies fiy [] = fig (x)), Vi.j. Hence the entire adjacency matrix satisfies (1) = A (2).
Meanwhile, deliberate mismatch in higher-order cumulants guarantees ( law (2),
Connection to Poisson Graphical Models. into the edge-rule template above (edges encode only E[X;X,l). Hence Theorem applies verbatim: they too are blind to high-order spike-count synchrony.
Pairwise log- linelr Poisson graphical models 99] fall
Constructive Example
Consider d = 3 binary processes, (f), ~ Bern(1/2), and define
(t) = X1 (t) @ X2(t) (XOR in {0, 1)).
(2)
All pairwise covariances vanish, yet the third-order cumulant
[[(2X; 1)(2X; 1)(2X, - 1)] = -1
K123
is non-zero. Any covariance-based network therefore returns an empty graph, missing the genuine ternary dependency.
D Proof of Theorem
We provide a self-contained argument that GCM can approximate any k-th order decision rule (A: 2 3) up arbitrarily small risk. All statements hold for finit f memend stochastic processes.
'The probability space of X(t) is assumed to have finite second moment, standard condition in neuro-signal modelling.
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D.1 Pre-liminaries and Notation
S (i1,-..,4 Ag) denotes the ROIs which the ground-truth rule depends.
A {0, 1J is the k-clique connecting S.
0% is the random graph generated by the Concrete reparameterisation ($ 3.1) at temperature
Each node v equipped with 2, = [X, (t), ea] E Rd, where ep is a unique, fixed ID embedding (one-hot random orthogonal vector).
X5 denotes the set of node features for the nodes in the set S, i.e., (zn/u E..
==max diammA (A*) (for a clique L=1).
D.2 Finite-Constane Approximation of the Oracle Graph
Lemma (Finite-K Concrete Approximation). For any > 0 and Ç >> 0 there exist a finite constant
K {6, parameters 6= (Ai E {+K}), and a temperature T < Tágg such that
Pr 93 A" 20
<
1-
(4)
Proof. Because the Gumbel noise Mij = = - log u) is almost surely finite, choose >> so that o ((K + max M2)/T) > - and a ((- + min M,,)/)) < with probability 1 0
D.3 Permutation-Invarian Injection via IDTagged DeepSets
Lemma 4 (ID-Tagged DeepSets Injection). Let Φ: td Rdd be injective and continuous. Define
h Σ o(z.), 2, (X,(t), ezl. DES
(5)
Then the mapping M {x9} + h is injective on all finite multisets of node features.
Proof. Distinct nodes carry distinct ID embeddings e,; hence (bz,,) + (2Zvg) whenever en * 02.
Sum aggregation therefore preserves multiset identity.
D.4 Truncated Universal Approximation
Lemma 5 (Probabilistic UA on Unbounded Domain). For any y) >> 0 there exist T > 0 and MLP pn such that
Pr[lix||>> T] <
(6)
(p,(h) (xx)) < vh 1 o(z)) |Xs||00 ST.
(7)
VES
Proof. Choose so that (6) holds (Markov's inequality suffices by finite variance). The image the compact cube [-T, 7 under the continuous sum is compact. Classical UA theorems guarantee an MLP on that attains
D.5 Main Theorem
Theorem 4 (Higher-Order Expressivity of GCM). Under the setting above, for any E 0 there exist parameters (6f.0F) temperature T5, and depth such that the GCM classifier satisfies
Pr[fer or (X(t) # v S
Proof. Allocate the error budget = Thrune with & = = Thrune e/3.
Adjacency Realisation. Apply Lemma 3 with (f, to obtain ($=,7*) such that adversarial graph mismatch occurs with probability
26

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One-Layer Aggregation. Run one message- messagg-passsig layer that computes h = Eves o(z,) (5) By Lemma h injectively encodes Xs.
Readout Approximation. Invoke Lemma with 7 to pick T and MLP Py satisfying (
Define
き = Pn Eves (Zz,) and set a final linear head so that g. o & matches the class labels.
Risk Upper Bound. Mis-classification can oceur from (i) wrong adjacency (5), (ii) UA enror (i)) or (iii) truncated tail mass (7/rune). Summation gives risk S { + *rrrnc
Remark 1. When & 2 the oracle graph reduces to single edge, and the above construction collapses conventional pairwise modelling. For k: 22 joint gradient updates over all A entries allow GCM to synthesise therbby escaping the impossibility bound for static pairwise networks established in $2.
E
Pseudo-code
In this section, we provide the pseudo-code the proposed framework GCM and BBA.
Algorithm GCM
Input:
Samples (A'), Level Mapping ø, Graph Numbers f, Learning Rate v/. Coefficient o, ,
Output: Predicted labels {V}, Learned graphs {9} Init (Á} ~ ~0, 1)). Other Trainable Parameters A while converge do for all X; in (X'} do
Compute A according to À = 0 A+M
Compute È for Á according to É 2 Ei=1 Ejet Aij
Binarize batch of À using E using BBA
Pair X; with G using o0-)
Compute Hidden LayersH GNN(N, 9.) Compute e, using ReadOut
Predict Label according to ÿJ softmax (WTer e; b)
Compute L1 according to L1 = E, log(Ui)
Compute L2
1 |P Pooll
Σ ((.j)) log of e,
|Pucal E(j,)
log
oTeec - Tal
Compute R according to A σ ( + AT)
(A
L= Li + a * Lg + 3 R
Update A by Å = A 1
Oc a AA
AA 0A A
Update 0 by = 0 E
BC 8Å
end for end while
Algorithm BBA
1: Input: A batch of adjacency matrices A E BBxNNN Thresholds k E R consists of the
expected edge numbers k(() of 6-th adjacency matrix A (b) e NNN in the batch
2: Output: Batch of network structures B
Transform (b) into q E N
3:
4: Construct M using m (b)
1
if (t
k00 1
(0)
0
otherwise.
5: Obtain Q by Q (b) = M M @
Q
B 6: Obtain { 1= by reshape Q to the shape of A
27

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F Revisiting the Role of Initial Structure
Table A2: Impact of Initial Structure on Classic Methods (Accuracy sia)
Method
PyntiCPasser
DyafiCPs
DynliCRa-ae
Cogfinte
SUIM
Inter
Int
Iocer
Iuter
Intra
Enter
Intza
OCN
0:774amazz
0.736as.0ss
0.425
1.55
1.60
0.545
2.290
0.50
0.552aam02
0,543
SAGE
0.29
a.798
0,46
0411aaas
0.833asuzz
0.508
0. 0.334
0.501 sares
0.657 a4.ma
0.596asuca
GAT
0.338a
0.212
0.202
0.542aaa
0.532ax.0s
0
0.905 parcs
06622
0 0,602aauus
a.495
0.44
0,572xxuus
0,308
0.290
0.534
1.465
0.612a
0.618
0:40
0,57 0.2
0. 23
0 0,269 20
0.567.3
SAGF
0.804gnerr1
0.63320
0.78320
0.656
0. 0:5
0. 0.328
2638
0.51
GAT
0:940.aaa21
6.847
0.82640
64552
0.804
0.649gnoer1
0.210an
0.6052 2.255
0. 0r5
GiN
D. g.145
0.152
0.422
0,545
0,476
0.238
0.247 20.00
0.529
0.528,
To highlight the limitations of pairwise methods, we first report our experiments conducted on GNNs in [Ab We compared the performance of directly inputting a Pearson- -based FBN versus using fully connected network (FCN) as input for the FBN. The results are marked with a red upward arrow to highlight instances where the predictive performance improved after the replacerme replacement. Half of the predictions showed varying degrees of improvement, demonstrating that Pearson-based FBNs, as a representative pairwise approach, fail to fully capture complex relationships in brain activity. Furthermore, using static FCNs revealed their limitations in capturing the structural information inherent in brain networks. These findings emphasize the need for FBN structure learning, which allows for the discovery of richer, more context-sensitive representations of brain activity. We conducted same experiments with GSL and BGI SOTAs for further validation, and the corresponding results are provided in Table A3 Again, over half of these SOTAs perform better without relying on original input structures, further underscoring the importance of exploring more reliable FBN construction techniques.
Table A3: Original vs. w/o Structure Settings (Accuracy+sa)
Mathol
DynHCPasay
Dy
Dyn
CoyStoc
SLIM
Inter
Iutra
Intra
Later
VIB GSL
0,900
0.925
0.772amt17
0,769
0,568 pares
0435
2112
0.79)
1.78
0,596
0.575anu55
0,318
0.313aa314
0.300
0,301
0.529
0.512_.azz
0,537anmt
0.522a000
HHGP-51
0.909gns:a
0,315anaai
0,647ane51
0.712
0.529
0.404
0.37
0.8223nm111
0,765
0.612anass
0.516gn
0.286
0.300
0.200
0.556amm
0.522
0,552an.00
0.504
1RGNN
0.308
1.842
0.804
0,768 20
g.707
0.397 0.397
0.545gn 2o6
0.591
0.591
0,343
0.318
0,
0,300
06666
0.68
01420
06672907
DOCL
a 68855
0.767 0.712
0.758
0.588aeam
0615
0,319
0,275 0.316
a.297
0.552
0.548_
0.51a
0.539
JC GNN
0.578asas 6.811
0.767
0.678anuss
0.653
0.33 aio0i
0.425
0851
0.602
0.321an
0.318
0,313
m642.
0658
166341
06644
BrainIB
0,941200:31
0.928
0.843_no01
0,839
a.819
0,800
0.453 21.1""
0.465
0.85.
0.60
2.662
0,346
0,332
0.333
0.325
0.6822mmp1
0644_a
0.640
0631asme
G Structure Learning Results Using Additional Metrics
Table A4: Comprehensive Performance Metrics (SEN, SPE, AUC)
Sy
Coggta
SEISS
TANY
Mecbol
520
L.91
0837
11811
B354
13:40
8.0%
BAD1
6.23
0.233
01355
E.5M0
11918
1166
1194
1374
6.00
1194
13-1
2.0%
11915
11999
0111
E.42
0.31
1550
0.11
14.5
LOV
0.594
0.50
INGS
0302
1720
0226
5.5
0.711
6216
510
M40
BARI
L/M
BALD
1.00
0.00
MALS
K20
05.6
516
0.644
195
3815
1676
100
0455
1009
0/55
0.00
176
BART
320
8.7
110
11966
U20
KAM
9AM
KTK
OCA
(5X)
MAI
020
6:20
0023
3/50
C.L.
1.5
1U25
064
EATE
506
15.9s UN0 AW9
390
1996
S/N
[591]
0M5
(849)
TOWS
(466
ETM)
28

---

To provide a more comprehensive analysis, we present the comparative results under various evalua- tion metrics, and the findings are consistent with those reported in the main text. Overall, our proposed GCM framework for four different resolutions rank at the top across the new metrics (SEN, SPE, and AUC), which further validates the effectiveness and robustness of our approach. The analysis again reveals that at the sample-level, although BNT shows exceptionally strong performance in intra-subject tasks, sampe demonstrates more comprchensive advantages in the more challenging inter-subject generalization tasks. Furthermore, at higher resolutions, GCMaabject and GCMgroup also exhibit outstanding performance, once again validating the value of our multi-resolution modeling framework.
H Sensitivity Analysis
(a)
Bample Lavel
Subject
Group Lavel
Project
Sample Lavel
Level
M00 sibeet subeet Acuuno
200
mateedte Intrandeert Kaurarr
(b)
(b)
Alwey
conder
Figure A1: Accuracy of GCM using different GNN backbones.
Figure A2: Accuracy GCM using different read out functions.
Choice of GNN Backhone The GNN module serves the of GCM, modeling local node interactions and extracting global representation of data. Figure A1 presents the performance of different GNN backbones. SAGE demonstrates superiority over other architectures under most of conditions, making it the preferred choice for our applications. Notably, further trials, such as using GCN on DynHCPActtintyy
Actimity the project-level, indicate potential for higher accuracy.
The readout function determines how node embeddings are aggregated into graph-level representa- tions. We evaluate the performance of various readout functions for GCM, as shown in Figure A2 MLP and Transformer are trainable, others are static. The results indicate that simple mean readout is effective for most conditions. In some cases, trainable functions like MLP or Transformer pro- vide improved precision, suggesting that the choice of readout function should align with specific application requirements.
Parameter Sensitivity Based on the training strategy, we further evaluate the prediction sensitivity to the hyperparameters and B. The results are shown in Figure [A3] In general, the performance of GCM is relatively stable to different choices of these two hyperparameters. However, the results also show that the predictability of GCM can be enhanced with proper coefficient set. For example, higher and moderate 3 show better predicting accuracy in group level conditions.
I Visualization of FBN Structures
We visualize the t0p-100 weighted edges in male and female FBNs learned by competitors. Subgraph- based methods are omitted because we focus on FBN structure learning for the entire brain rather than sub-FBN extraction. Nodes are represented as colored segments along the circle, edges as curves, and segment widths indicating node degrees. Because our aim to extract high-order patterns from MTS, and the co-activated brain regions are depicted in the main paper as Fibure 5 comparison on edge and node distributions of the learned FBNs between baselines. According to the visualizations, the averaged FBN structures of baselines represent more randomness compared the
we mainly present the
29

---

96
Accuracy
94
92
70.0
64
65.0
Sample-level
62.5
Subject-level
Group-level
60.0
Project levell
value of a
Figure A3: Results using different values a and 8
female DOGL
female HGP-SL
VIB-GSL
male IBGNNN
male IGS
male DCGL
male HGP-SL
male VIB-GSL
male GCM
Figure A4: Male (red) and female (blue) FBNs on DynHCP Gender show top 100 weighted edges obtained by different methods.
structure learned by GCMgroup, emphasizing the inadequacy of simple averaging across research levels. Prominent hub nodes can be observed in the FBNs learned by GCMgroup, further enhancing the result that high-order interactions are not randomized [17]. Notably, GCMgroup reveals distinct patterns: female FBNs have more nodes and fewer hubs than male FBNs, which aligns with the conclusions of the previous study in neuroscience 194) demonstrating that the learned structures are both meaningful and non-trivial.
Reproduction Settings
The detailed settings of our experiment provided in this section.
J.1 Parameter Settings
For the traditional methods, they are implemented using the scikit-leary package with their default parameter settings. For the method needing input structures, we computed graph structures by selecting the t0p-10% entries of the Pearson correlation matrices calculated from samples. We also set scenario where the input structures are fully connected networks, representing the same condition our proposed GCM where there initial structures. We randomly selected 10
30

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samples for each subject on each data set for speeding up, except for SLIM. We shuffled and split each data set into training/vaiidatioonness sets according to each task by the ratio of 7 2.
We set r = 1 in the Gumbel-softmax, Ta = in the contrastive loss, a and / are selected by grid searching. SAGE is set as the backbone for GCM in comparison. Common parameter settings summarized as follows:
Learning Rate is as 0.001 for each competitors.
Epochs are to 000 for each competitors.
Early Stop thresholds are set to 10 for deep models.
Batch Size is set to 32 for deep models.
Layers are set as 2 for deep models.
Hidden Dimensions are set as 128 for deep models.
Output Dimensions are set as 2 for deep models.
Backbone is set for GSL and BGS models default: GIN for VIB-GSL, GCN for HGP-SL,
IBGNN and IGS, respectively.
Readout is set as Mean pool in GNN-based methods for fair comparison except for HGP-SL, which is set HGPSL-Pool by default.
Head is set to 8 for GAT.
Network Density is set to 10% for methods needing a fixed threshold.
The rest the parameters are set as default.
J.2
Environment
All the experiments were performed on Linux cluster equipped with Intel(R) Xeon(R) Platinum 8358P CPU @ 2.60GHz, Nvidia A800 80G GPU and 520 GB RAM.
J.3 Data Description and Preprocessing
Human Connectome Project (HCP) The Human Connectome Project offers valuable, publicly accessible neuroimaging dataset that encompasses comprehensive array of imaging, behavioural, and cognitive data. Our study leverages dynamic networks derived from both resting-state and seven distinet task-based fMRI paradigms. Thus, there are three distinct predicting objectives according to the dataset, namely Activity, Age, and Gender, respectively. We use open-source version standardized and processed by NeuroGraph The dynamic graphs are generated via sliding window approach, characterized as window length of 50, stride of 3, and dynamic length of 150, which is applied to the preprocessed time series.
Cognitive State Cognitive State dataset (Cog State) collects EEG data from a group of 60 healthy undergraduate subjects, resulting in total of 900 records. The primary objective is to validate the reproducibility of EEG metrics across different cognitive states. The EEG records categorized into five cognitive states: Eyes Open (EO), Eyes Closed (EC), Math (Ma), Memory (Me), and Music (Mu). Each EEG record, originally sampled resolution of 500 Hz. comprises substantial 150, 000 data points across 61 channels. These channels adhere to the extended 10-20 system, a widely recognized standard in EEG studies. We initially down-sampled the data to 100 Hz using EEGLAE comprehensive MATLAD plug-in designed for EBG data analysis. The down-sampling process resulted in manageable 61 30, 000 matrix per record. Subsequently, employed non overlapping sliding window technique to segment the data into manageable 3-second fragments. This segmentation yielded approximately 100 fragments per subject, each represented as a 61 x 300 matrix and retained its corresponding cognitive state label.
period of 5 minutes, thus
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The SLIM dataset includes stst-state fMRI data from cohort of 1, 045 subjects, using gender as the labels for analysis. We screened out the incomplete data and retained 541 subjects. We constructed brain functional networks based on the Power 264 atlas, well-established brain parcellation template. We remove 28 undefined regions resulting 236 ROIs as the nodes of FBN. The methodology for this construction is twofold:
SLIM
a MATLAB plug-in, we delineate brain regions according to Power's cortical parcellation template, identifying 264 regions. The time series signal for each region is derived by averaging voxel signals within a 5mm radius sphere.
Using DPABI 10
2. A 30-second, con-overlapping sliding window technique applied to segment time series into fragments (dimensions: 236 x 30). Pearson correlations between brain regions within each window are calculated, forming dynamic connectivity matrices (236 236).
K Introduction of Baselines
Table A5: Taxonomy of Baselines
Characteristic
Traditional
GNN
BGI
GSL
GCM
End-to-End
X
d
Structure Learning
X
Global Constraint
X
X
X
Multi Resolution
X
X
X
Adaptive Threshold
X
X
X
To verify the suitability of GCM on FBN learning, choose the following baselines. Property comparison between GCM and baselines are summarized in Tabel
Traditional Models To ensure fair assessment, these methods implemented using scikit-learg package
LR:
logistic regression, using logistic function model the relationship between features and the target variable.
SVM: Support vector machine, which determines the decision plane through support vectors. We use RBF kernel for implementation.
RF: Random Forest, ensemble method that operates by constructing a multitude of decision trees and outputs the mode of their predictions.
XGBoost: Extreme Gradient Boosting, an efficient implementation of gradient boosting trees that builds models in sequential, stage-wise fashion.
MLP: feed-forward artificial neural network, features multiple layers of interconnected 'neuron'.
GNN Models To ensure a fair assessment, these models are implemented in standardized dense form GCM using PyTorch Geometric
GCN 74
graph convolutional network, adapts convolutional processes to graph structures.
SAGE 79]: an inductive learning framework that creates node embeddings by sampling and aggregating features from the nodes' local neighbourhoods.
GAT [80]: graph attention network, utilizes self-attention mechanisms to assign varying degrees of importance to different nodes in a graph.
GIN 8 graph isomorphic network, aims to match the distinguishing power of the Weisfeiler-Lehman graph isomorphism test.
index.inexx.. html
cffri.org/DPii.
scikit-learn.c. org
pptorch- www.://mtiii...atteeeoooooooo
32

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GSL SOTAs
The open-souree methods are experimented with using the original code released by their authors.
VIB-GSL 70 framework employing the information bottleneck principle to selectively filter out irrelevant information thus enhancing graph representation learning.
combines structure learning with hierarchical graph pooling in GNN to capture and preserve vital topological information from graphs
HGP-SL [
learn FBNs based on pair-wise interactions between features learned by the encoder designed for DTI data. We re-implement it according the original paper suit EEG and fMRI data.
DGCL 88:
BGI SOTAs These methods are experimented with using the original code released by their authors.
IBGNN 82]: constructs FBNs by Pearson correlation and utilizes a two-layer MLP to merge topological information for prediction.
IBGNN+
82
learns a sparsified mask unified filter the FBNs after IBGNN is
trained.
IGS [1 learns to iteratively remove task-irrelevant edges of FBNs during training for sparsification.
IC-GNN [53]: A Granger causality-nnspired graph neural network using subgraph for interpretable brain network-based psychiatric diagnosis.
BrainIB 39|: A mutual information-base framework for interpretable and robust brain network analysis in psychiatric diagnosis.
These models were incorporated suggested during the review process to further benchmark GCM's performance.
Additional SOTA Models
BNT 33
Brain Network Transformer, state-of-the-ar model that applies a Transformer architecture to brain networks, using attention mechanisms to achieve high predictive performance.
Hybrid 49): A conceptually related model that learns a fixed quantity of explicit high-order interactions (hyperedges) in an end-to-end manner.
L Limitations
The limitations of GCM framework are summarized follow: (1) Though the framework is generalizable to other MTS-based tasks, further validation on diverse modalities (c.g., MEG) and clinical conditions is needed. (2) Although GCM captures high-order synchronization and yields interpretable results, the underlying neurobiological mechanisms remain underexplored. (3) While formalize the semantic distinctions across four modeling resolutions, their joint modeling and integration remain unaddressed. (4) The experiments in this study are conducted on datasets with relatively balanced class distributions. The performance of the contrastive loss, particular, on highly imbalanced clinical datasets is an important area for future investigation.
M Societal Impacts
Given that the research this paper involves neurological analysis, it could used as an approach to discase diagnosis. Thus, it is essential to declare the potential negative societal impacts of this study, even though is currently the research phase and has not yet been applied in practice. Specifically, in the process of Al-assisted discase diagnosis, erroncous results are inevitable. Such errors have severe consequences for patients and society. Therefore, in real-world medical diagnostic scenarios, the final decision should always rest with the physician's 's diagnosis.
33


Reviews:
[{"review_id": "op2uM0KRIt", "reviewer": "Reviewer_pEum", "summary": "This paper introduces the GCM framework to learn sparse, binary, high-order functional brain networks end-to-end. It addresses limitations of pairwise models by combining supervised label loss, contrastive regularization, and a sparsity prior within a multiresolution hierarchy (sample, subject, group, project). The framework employs Gumbel–Sigmoid relaxation alongside a batch binary algorithm. Experiments show a boost in classification accuracy and reduction in computation time across five datasets.", "questions": "1. How are hyperparameters chosen?\n\n2. Can you Show the relationship between different levels need more motivation, what relationship are extracted, are there any difference between different or different methods. So we can get more motivation about why you propose the four-level hypergraph structure.\n\n3. Since traditional ml methods can go great in performance, what is the significance of using these DL-based meethods? They acuique high computation resource.\n\n4. How is overfitting controlled?  Networks learned at the group and project resolutions exhibit a tendency to overfit due to label leakage, suggesting that the current regularization may be insufficient for tasks with strong population-level supervision .\n\n5. Is contrastive regularization robust on imbalanced datasets? The contrastive regularization assumes roughly balanced classes to learn meaningful within-class similarity; the paper does not report experiments on imbalanced datasets, leaving its robustness in such settings untested.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The paper delivers a formal proof showing that traditional pairwise FBN models cannot capture high-order statistical dependencies, providing a solid theoretical foundation for its high-order approach.\n2. Experiments are extensive and results comprehensive, demonstrating the framework’s effectiveness across multiple datasets.\n3. The multiresolution hierarchy (sample, subject, group, project) is well-conceived, enabling the model to capture interactions at various scales and offering interpretable insights from individual time windows to population-level patterns.\n4. Overall clarity and quality are high, with methods and results presented in an accessible manner.\n\n**Weaknesses:**\n\n1. The motivation for relationships across hierarchy levels could be elaborated further to justify, for example, differences between relationships extracted at the sample, subject, group, and project levels, and how these differ from relationships captured by baseline (pairwise) methods.\n2. Additional evaluation metrics (beyond accuracy) and repeated experiments would further validate results and guard against overfitting or dataset-specific artifacts. Hyperparameter tuning details are not reported; including these would strengthen confidence in the model’s robustness.\n3. Clinical translation is limited: although classification accuracy improves, there is no evaluation on real-world clinical cohorts or tasks directly tied to patient outcomes, which is important for demonstrating practical utility.\n4. By enforcing a binary adjacency matrix, the framework omits edge weight information that may reflect graded coupling strengths, potentially missing subtle but meaningful connectivity variations.", "was_revised": true, "final_justification": "I thank the authors for the rebuttal. It has addressed most of my concerns and I have raised my score to 5.", "rebuttal": "Thank you for your thoughtful and constructive comments! Your feed back on the presentation issue allows us to make it more understandable to readers with different backgound.\n\nBelow, we address each question in turn and clarify the points raised. If our paper is eventually accepted by the conference, we will implement the responses regarding the clarity of the paper to make it easier to follow by readers.\n\n# Responses to Questions\n## Q1. How are hyperparameters chosen?\n\nFor fair comparison, most hyperparameters followed standard practices or baseline defaults (details in Appendix H.1). Key coefficients, α and β, are tuned via grid search, with the results presented in our sensitivity analysis (Appendix F, Fig. A3). For network sparsity, after observing that performance plateaus at around 5% density (Table 4), we adopted a 10% threshold for baselines requiring a fixed density to ensure a consistent and robust evaluation. Furthermore, extensive sensitivity and ablation studies (Sec 6.3, Appendix F) confirm that GCM's performance is robust across various settings and to its core components.\n\n## Q2. The motivation behind the four-level hierarchy.\n\nThank you for this essential question! We note the reviewer's valuable point on **clinical translation**. To make the distinction clearer and more concrete, let us illustrate using a common clinical task: classifying patients with **Major Depressive Disorder (MDD)** from **Healthy Controls (HC)** using fMRI data. In a typical fMRI analysis pipeline, the continuous brain signal is first segmented into 'samples' using a sliding-window approach.\n\nFor each level of analysis, GCM learns a different type of information, a task that traditional pairwise methods struggle with due to their decoupled and heuristic nature.\n\n* **Sample-level:** GCM learns a transient FBN from a single sample to capture momentary brain states.\n    * **How traditional methods fail:** Traditional methods rely on a strong prior assumption of **pairwise relationships** (e.g., correlation), which is insufficient for capturing higher-order dependencies  (**Theorem 1**). For example, consider a triadic relationship where a brain region (C) activates only when its two driving regions (A and B) are in **opposite states** (i.e., one is active and the other is inactive). This triadic interaction can be directly modeled as a high-order XOR relationship. However, pairwise statistics would fail to capture it, as no stable pairwise synchrony exists between any of the three regions. \n\n* **Subject-level:** GCM learns a single, stable \"neural fingerprint\" for each individual (e.g., \"Patient A\"). This graph is optimized end-to-end from all of Patient A's data to best represent their unique traits for MDD classification.\n    * **How traditional methods fail:** They cannot learn this directly. They must resort to heuristics, such as: 1) averaging all of Patient A's raw time-series data first, then computing one correlation graph (destroying dynamic information), or 2) creating many sample-level graphs and then averaging the resulting connectivity matrices (losing structural specificity). Neither approach optimizes the graph for the actual classification task.\n\n* **Group-level:** GCM directly learns a prototypical FBN representing the canonical biomarker for the entire MDD cohort, and another for the HC cohort.\n    * **How traditional methods fail:** They cannot learn such a graph directly. They must first generate subject-level graphs and then rely on separate, post-hoc statistical analyses (e.g., performing t-tests on every single edge) to identify a \"significant\" group pattern. This decouples network construction from the learning objective.\n\n* **Project-level:** GCM learns a single FBN that models the key structural differences between the MDD and HC populations using the entire dataset. It holistically discovers a unified network where the most discriminative pathways are identified from a global perspective.\n    * **How traditional methods fail:** They would again rely on post-hoc filtering, such as selecting edges based on the statistical significance of their weight differences between the MDD and HC groups.\n\nIn summary, our core contribution is replacing these decoupled, heuristic, and often local (edge-wise) construction steps with a **single, globally-aware, end-to-end optimization framework**. This allows GCM to learn task-relevant, high-order network structures that are inaccessible to traditional pairwise models.\n\n## Q3. Value of using deep learning–based methods?\n\nThank you for this critical question! While traditional ML can perform well, it relies on a **two-stage, decoupled process**: constructing FBNs with predefined pairwise heuristics (e.g., Pearson correlation), then classification.\n\nCrucially, many existing DL methods inherit this limitation. In contrast, **GCM introduces a paradigm shift**: it learns the FBN structure end-to-end, optimizing for task-relevant, high-order dependencies without such presumptions. The resulting performance improvements and novel structural insights (Table 2, Fig. 5) justify our data-driven approach. We will clarify this motivation in the revision.\n\n## Q4. How is overfitting controlled in group-level?\n\nThank you for proposing this is excellent and critical point! You are correct that learning a class-specific FBN could risk label leakage if not properly regularized. Our framework incorporates two key mechanisms to explicitly control for this:\n\n1. **Targeted Regularization via Global Constraints**: Our proposed global constraints act as crucial regularizers. While the performance impact varies across different tasks and resolutions, our ablation study (Fig. 4) reveals a critical trend that supports our claim. The performance degradation from removing these constraints is most pronounced precisely in the settings you highlighted: **the group and project levels, particularly for inter-subject prediction**. For example, removing the subject consistency loss (L2​) or the sparsity prior (R) leads to a clear performance drop in these higher-level resolution tasks. This demonstrates that our constraints are actively working to prevent the model from collapsing to trivial solutions, especially where the risk of overfitting is highest.\n    \n2. **A Principled Reinterpretation of the Modeling Goal**: We propose a conceptual shift for group-level analysis. The goal is not to learn a universally \"correct\" FBN, but to discover a **prototypical graph structure that is maximally discriminative for a given class** (e.g., a canonical network for the \"female\" group). This learned structure acts as a task-optimized computational graph. Its validity is then confirmed by its generalization performance on unseen test data. The high, but not perfect, accuracy in **Table 3** indicates that GCM has learned generalizable class-specific biomarkers, not just overfitted to the training data.\n\nFurthermore, the interpretability of these learned prototypes **(Sec 6.4 and Appendix G)**, aligns with established neuroscience findings, providing external validation that the model has captured meaningful biological patterns rather than simply memorizing noise.\n\n## Q5. Adaptation to imbalanced dataset.\n\nThank you for raising this practical and forward-looking question. In our current experiments, we follow the standard setting in much of the neuroscience literature where datasets are curated to maintain relatively balanced class distributions, which ensures that the contrastive regularization operates under stable conditions. We fully agree that explicitly designing contrastive regularization for imbalanced conditions is an essential next step for broader clinical application. We will include this as a key direction for future work in our revised discussion section.\n\n# Response to Additional Weaknesses Not Covered in the Questions Section\n\n## [W1] Clinical translation is limited\n\nThank you for highlighting the important gap between methodological innovation and clinical deployment. We agree that demonstrating utility on clinical cohorts is a crucial next step.\n\nIn this foundational work, our primary objective is to establish a robust and interpretable conceptual framework for learning high-order FBNs across multiple resolutions. We view this as a necessary prerequisite for reliable downstream clinical applications. To this end, we deliberately focused on building a general-purpose learning framework rather than tailoring it to a specific disorder.\n\nThat said, we are currently adapting the GCM framework to a MDD fMRI dataset. This follow-up study aims to assess the reproducibility and interpretability of the learned network biomarkers in a disease-specific setting. We will mention this ongoing effort and explicitly acknowledge the clinical limitations of the current version in the revision.\n\n## [W2] Binarized adjacency omits edge weight information. Possibility of missing weak connections.\n\nThank you for this insightful comment. The choice between binary and weighted graphs is indeed a critical design decision with important trade-offs.\n\nIn this work, we chose to learn binary graphs to prioritize identifying a sparse, interpretable **scaffold** of the most salient high-order connections for a given task. This aligns with the goals of many Brain Graph Interpretation (BGI) methods and forces the hard decisions about which connections are truly critical, rather than retaining many weak, potentially noisy edges.\n\nHowever, our GCM framework is **highly flexible** and not limited to binary graphs. As you suggest, it can be seamlessly adapted to learn weighted FBNs. This can be achieved by simply removing the Batch Binarization Algorithm (BBA) step and using the continuous adjacency matrix directly in the GNN. The end-to-end optimization process remains the same.\n\nWe agree this is an important aspect to clarify. We will add a discussion on this extension and the binary/weighted trade-off to the revised manuscript."}, {"review_id": "nW2meJOn2s", "reviewer": "Reviewer_nXo8", "summary": "This paper extracts high-order FBN structures under global constraints, implemented as a Global Constraints-oriented Multi-resolution (GCM) FBN structure learning framework. This framework integrates three types of global constraints (expected edge numbers, data sources, and data labels) to support FBN structure learning across four distinct modeling resolution levels (sample, subject, group, and project), with experimental results showing GCM achieves higher relative accuracy and less computational time.", "questions": "In Figure 1, the local constraints and global constraints are not clearly explained, which seriously undermines the expression of the research motivation.\n\nAs one type of global constraint, what does \"data source\" mean? Does it refer to \"dataset\"? According to the presentation in the experimental section, the 5 datasets are trained separately, so it seems unreasonable for \"data source\" to serve as a constraint.\n\nFigure 2 presents the Sample-Level Structure Learning Results. Figure 3 shows the Multi-Resolution Structure Learning Results, where \"Multi-Resolution\" includes the Subject, Group, and Project Levels. Therefore, the four distinct levels do not work collectively, and their combination appears incomplete.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "This work is relatively clearly expressed. To address the limitation of current FBN models in capturing high-order dependencies, it incorporates three types of global constraints (expected edge numbers, data sources, and data labels) to enable the learning of FBN structures across four distinct levels of modeling resolution (sample, subject, group, and project). A large number of experiments are conducted on 5 datasets and under 2 task settings, with comparisons made against 7 baselines and 8 state-of-the-art methods. However, some expressions are not clear enough: for example, Figure 1, as well as the relationships between the three types of global constraints and between the four distinct levels, are not clear.", "was_revised": true, "final_justification": "Most of my concerns have been well answered by the author. I choose to maintain the original score.", "rebuttal": "We sincerely thank you for the thoughtful and constructive feedback! As our paper uses deep learning to solve neuroscience problems, clear definitions are paramount. Your comments on expression issues are invaluable for helping us polish the paper for a broader audience. We respond to each concern below and will incorporate these clarifications into the final manuscript.\n\n## Q1: Clarifying local vs. global constraints in Figure 1\n\nThank you for pointing out this critical ambiguity, which is central to our motivation. We apologize for the lack of clarity and will significantly revise **Figure 1** and its caption to make the distinction intuitive.\n\nThe core difference is the **scope of influence**:\n\n- **Local Constraints** operate on a **node-pair basis**. Traditional methods that calculate Pearson correlation between two brain regions to determine a single edge weight are using a local constraint. The final graph is an aggregation of many such independent, local decisions.\n    \n- **Global Constraints**, in contrast, operate on the **entire graph structure** at once. In our framework, these are:\n    \n    1. **Task Labels**: A supervised signal that shapes the entire network to be discriminative for the classification task.\n\n    2. **Learnable Sparsity Prior**: The adaptive \"Expected Edge Number\" that dictates the overall sparsity of the whole network.\n\n    3. **Intrinsic Data Constraints** (\"Data Source\" in the submission): This is the most fundamental constraint, referring to guidance originating directly from the input data itself. It has two key components:\n\n        a) **Subject-Sample Hierarchy**: We use the subject identity of each sample to apply a contrastive loss, enforcing a consistent \"neural fingerprint\" across all graphs from the same person.\n\n        b) **Raw Signal Dynamics**: Most crucially, our end-to-end model is directly constrained by the high-order temporal synchronization patterns embedded within the raw multivariate time series (MTS) itself. This is the ultimate source material for the FBN.\n        \nThese global forces guide the learning of the entire FBN as a single entity, rather than building it piece by piece. We will update Figure 1 to visually represent this \"whole-graph\" vs. \"edge-by-edge\" distinction.\n\n## Q2: What does “data source” mean as a global constraint?\n\nThank you for this important question. You are correct that using the term “data source” was ambiguous, and we apologize for the confusion. Our intention was to encompass two distinct and fundamental types of constraints that originate **directly from the input data itself**, prior to considering task labels or model-based priors.\n\n1. **First, the data's organizational structure**: This refers to the **subject-sample hierarchy**. Knowing that multiple data samples belong to a single individual allows us to apply a powerful constraint via the contrastive loss (L2​), guiding the model to learn a consistent and stable \"neural fingerprint\" for each person.\n    \n2. **Second, the data's intrinsic signal properties**: More fundamentally, this refers to the constraint imposed by the **raw multivariate time series (MTS) itself** (i.e., the fMRI or EEG signals). Because our GCM framework learns the FBN structure **end-to-end**, the learned graph is inherently constrained by the complex co-activation patterns and high-order temporal dynamics within the raw signals. This is a core advantage over traditional two-stage methods, which first abstract the data into a (often pairwise) correlation matrix, thereby losing the richer information embedded in the original signal dynamics that our model directly leverages.\n    \nWe realize that bundling these two powerful but distinct ideas under the single, ambiguous term \"data source\" was a mistake and obscured our contribution.\n\nTo rectify this, we will revise the manuscript to treat these concepts separately. We will explicitly define the **subject identity** as a key global constraint for regularization. We will then move the discussion of the **intrinsic MTS signal constraint** to our core motivation in the Introduction and Methodology, highlighting it as the foundational reason for adopting an end-to-end, high-order learning paradigm.\n\nThis separation will clarify the dual roles of the data: (1) providing organizational structure (subjects) and (2) providing raw dynamic content (MTS). Thank you for pushing us to articulate this crucial aspect of our framework.\n\n## Q3: Clarifying the meaning and purpose of multi-resolution design\n\nThank you for raising this important point. You are correct, the relationship between the four modeling resolutions was not made sufficiently clear. **The core relationship we aim to establish is one of deliberate methodological and semantic separation.** You are also correct in observing that our framework handles these levels **separately**, not as a single, jointly-trained model. We apologize if our use of \"Multi-resolution\" suggested a data fusion or integration approach.\n\nOur central argument is that this separation is a critical and necessary contribution. To make the value of this methodological separation concrete, let us illustrate using a clinical task which stems from our ongoing work: classifying patients with **Major Depressive Disorder (MDD)** from **Healthy Controls (HC)** using fMRI data. In a typical fMRI analysis pipeline, the continuous brain signal is first segmented into 'samples' using a sliding-window approach.\n\nFor each level of analysis, GCM learns a different type of information, a task that traditional pairwise methods struggle with due to their decoupled and heuristic nature.\n\n* **Sample-level:** GCM learns a transient FBN from a single sample to capture momentary brain states.\n    * **How traditional methods fail:** Traditional methods rely on a strong prior assumption of **pairwise relationships** (e.g., correlation), which is insufficient for capturing higher-order dependencies  (**Theorem 1**). For example, consider a triadic relationship where a brain region (C) activates only when its two driving regions (A and B) are in **opposite states** (i.e., one is active and the other is inactive). This triadic interaction can be directly modeled as a high-order XOR relationship. However, pairwise statistics would fail to capture it, as no stable pairwise synchrony exists between any of the three regions. \n\n* **Subject-level:** GCM learns a single, stable \"neural fingerprint\" for each individual (e.g., \"Patient A\"). This graph is optimized end-to-end from all of Patient A's data to best represent their unique traits for MDD classification.\n    * **How traditional methods fail:** They cannot learn this directly. They must resort to heuristics, such as: 1) averaging all of Patient A's raw time-series data first, then computing one correlation graph (destroying dynamic information), or 2) creating many sample-level graphs and then averaging the resulting connectivity matrices (losing structural specificity). Neither approach optimizes the graph for the actual classification task.\n\n* **Group-level:** GCM directly learns a prototypical FBN representing the canonical biomarker for the entire MDD cohort, and another for the HC cohort.\n    * **How traditional methods fail:** They cannot learn such a graph directly. They must first generate subject-level graphs and then rely on separate, post-hoc statistical analyses (e.g., performing t-tests on every single edge) to identify a \"significant\" group pattern. This decouples network construction from the learning objective.\n\n* **Project-level:** GCM learns a single FBN that models the key structural differences between the MDD and HC populations using the entire dataset. It holistically discovers a unified network where the most discriminative pathways are identified from a global perspective.\n    * **How traditional methods fail:** They would again rely on post-hoc filtering, such as selecting edges based on the statistical significance of their weight differences between the MDD and HC groups.\n\nIn summary, our core contribution is replacing these decoupled, heuristic, and often local (edge-wise) construction steps with a **single, globally-aware, end-to-end optimization framework**. This allows GCM to learn task-relevant, high-order network structures that are inaccessible to traditional pairwise models. We believe establishing this clear, principled foundation is a necessary prerequisite before any meaningful cross-level integration can be attempted in future work."}, {"review_id": "ZrXUIJ1Q9o", "reviewer": "Reviewer_eH9B", "summary": "The manuscript proposes GCM, that effectively captures high-order interactions of brain regions by dynamically learn the structure and parameters under both local and global constraints. The manuscripts provides both theoretical analysis and empriical demonstations.", "questions": "1. In appendix C, you assign each node with an identity encoding. Does that mean your settings are always transductive, i.e., the nodes remain the same, while only their features are changing?\n\n2. I did not find definition of x_S and M in Appendix C. Could you make it clearer? Also, I suggest to make your proof in appendix more self-contained. Currently some definitions of notations are unclear to me and it is not easy to scroll back and forth constantly between the main text and the appendix, which makes it hard to follow your proof.\n\n3. Does the high-level modelling help the low-level tasks. For example, on sample-level tasks, do you have an ablation that removes the higher-level graph modelling (group and project level) and see how the performance shifts?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nThe proposed method shows superior performances on different datasets with different resolutions, demonstrating the effectiveness.\n\n\nWeakness:\n\n1. The term \"global constraints\" is introduced in the introduction without sufficient explanation, making it difficult to follow. The authors should consider providing a clearer and more intuitive description early on to help readers understand its significance and role in the paper.\n\n2. The proposed method contains many components. However, it is unclear that which components contribute to the modelling of high-order interactions.\n\n3. While ablation studies of BBA are presented in Section 6.3 and Table 4, both the table caption and the accompanying explanations in the main text are unclear, making the results difficult to interpret. For instance, it is not explicitly stated which columns correspond to the \"fixed threshold\" variant and which to the \"adaptive\" one. Additionally, the table lacks an ablation of a non-discretized version of the method, which would help clarify the contribution of the discretization step.\n\n4. Missing some important baseslines or discussions about some related works, such as [1][2].\n\n\n[1] Qiu, W., Chu, H., Wang, S., Zuo, H., Li, X., Zhao, Y., & Ying, R. (2023). Learning high-order relationships of brain regions. arXiv preprint arXiv:2312.02203.\n\n[2] Kan, X., Dai, W., Cui, H., Zhang, Z., Guo, Y., & Yang, C. (2022). Brain network transformer. Advances in Neural Information Processing Systems, 35, 25586-25599.", "was_revised": true, "final_justification": "The work has some merits: First, it shows the effectiveness at multiple resolutions. Second, it provides some theoretical analysis of high-order dependencies.\n\nHowever, my major concern is that the proposed method actually is not closely related to high-order dependencies. As pointed out in the discussion with the author, the proposed method still model those high-order depedencies implicitly in learned parameters (it's also questionable to call them high-order dependencies). The explicit structure is not provided, while only the projection (pairwise edges) is left. In that case, the method shows no more advantage than a transformer model [2], and less intrepretability than explicit models [1].\n\n\n[1] Qiu, W., Chu, H., Wang, S., Zuo, H., Li, X., Zhao, Y., & Ying, R. (2023). Learning high-order relationships of brain regions. arXiv preprint arXiv:2312.02203.\n\n[2] Kan, X., Dai, W., Cui, H., Zhang, Z., Guo, Y., & Yang, C. (2022). Brain network transformer. Advances in Neural Information Processing Systems, 35, 25586-25599.", "rebuttal": "Thank you for your insightful and detailed review. Your feedback on clarity is especially valuable, as our work introduces a paradigm shift in applying deep learning to functional brain network (FBN) modeling. You have provided us with an excellent opportunity to better articulate our contributions. We are confident that the following responses and the resulting revisions will make the novelty and significance of our work much more apparent.\n\nBelow, we provide point-to-point responses to each of your questions and concerns.\n\n# Responses to Questions\n## Q1: Transductive vs. inductive setting\n\nThank you for raising this important point. Our proposed GCM framework **does support inductive learning**, as it is compatible with GraphSAGE as an optional backbone, which is originally developed for inductive representation learning on graphs. However, in our theoretical proof in Appendix C, we focused on a transductive setting for the following reasons:\n\n1. Assigning fixed nodes is more convenient for notation in the proof. The conclusion regarding the approximation of k-th order relationships still holds for a changeable node set. We will clarify this extension in the revised version.\n    \n2. This choice is grounded in our **application domain**. In fMRI/EEG analysis, the node set (i.e., brain regions) is fixed by a standard atlas for all subjects. The transductive setting therefore directly mirrors this real-world use case.\n    \nNonetheless, we agree that extending GCM to inductive settings is a promising direction for future work and will add this discussion to the paper.\n\n## Q2: Notation issues in Appendix C\n\nThank you for your careful reading and detailed feedback on Appendix. We sincerely apologize for the typographical issues and lack of clarity in the notation. Specifically:\n\n- The symbol $\\mathbf{x}_S$ denotes the set of node features for the nodes in the set $S$, i.e., {$\\mathbf{z}_v|v\\in S$}.\n\n- The matrix $M$ refers to the Gumbel noise matrix $\\mathbf{M}$, as defined in the main text’s description of the sampling process (originally near Eq. 5).\n\nIn the revision, we will edit the Appendix to be self-contained by defining all variables upon their first appearance and minimizing references to the main text.\n\n## Q3: Does high-level modeling help low-level tasks?\n\nThank you for this insightful question, as it touches upon the foundational philosophy of our work. The primary goal of this paper is to **first establish that the 4 modeling resolutions are semantically distinct and non-interchangeable**. This is a critical point that is often overlooked in prior FBN literature. Our framework (GCM) is therefore designed to learn FBNs at each level _individually_.\n\nThis approach represents a deliberate **paradigm shift** in how machine learning is applied to computational neuroscience, where respecting the domain's fundamental constructs is a prerequisite for generating scientifically meaningful results. Our deliberate separation of the resolutions is crucial for two main, concrete reasons:\n\n1. **Interpretability**: In neuroscience, different resolutions represent **different scientific constructs** (e.g., a fleeting mental state vs. a stable personal trait). Blending them might mix information across these levels and lead to scientifically uninterpretable results. Our framework’s separation ensures that a \"subject-level\" FBN truly reflects an individual's stable trait.\n    \n2. **\\Principled Foundation**: Before one can explore how these levels should interact, it is necessary to first prove they are distinct and then be able to model each one robustly. Our Theorem 3 (Appendix A) provides the formal proof for this non-interchangeability. By demonstrating GCM's strong performance at each isolated level, we pave the way for future work on multi-resolution integration.\n    \nTherefore, evaluating each resolution individually, as the reviewer suggests ablating, is exactly how our main experiments were designed. We will revise the introduction and methodology to make this design choice and its motivation more explicit.\n\n# Response to Additional Weaknesses Not Covered in the Questions Section\n\n## [W1] Lack of explanation for term \"global constraints\" in Introduction\nThank you for pointing out this ambiguity. We will revise the introduction to provide a clear and intuitive explanation immediately after the concept is introduced. To ensure clarity, we will add the following definition:\n\n> \"In our framework, **local constraints** refer to priors that guide the formation of individual edges based on pairwise statistics (e.g., correlation). In contrast, **global constraints** are top-down priors that shape the entire network structure holistically. These include task labels, subject identity, and an adaptive sparsity, which guide the optimization of the adjacency matrix as a single entity rather than a collection of independent edges.\"\n\nWe will also revise **Figure 1** and its caption to visually and textually reinforce this distinction.\n\n## [W2] Modeling components responsible for high-order interactions\n\nThank you for this important question. The modeling of high-order interactions is primarily achieved through the **synergy between our prototype-based graph generator and the Gumbel-Sigmoid relaxation mechanism**. In this module, the entire adjacency matrix is treated as a single, holistic, learnable object. Gradients from the task objective flow back through the GNN and the Gumbel-Sigmoid sampler to update the underlying continuous prototype matrix. This means the update for any single edge is conditioned on the state of the entire graph, allowing the model to learn complex, high-order dependencies that cannot be decomposed into simple pairwise interactions. This holistic optimization directly addresses the limitations of pairwise models, which, as we formally prove in Section 4, are inherently blind to such dependencies.\n\nThis holistic learning mechanism is the cornerstone of our design, chosen specifically to overcome the primary limitation of traditional methods: their inherent blindness to high-order interactions. Conventional pipelines build networks using pairwise measures (e.g., correlation) as a fixed pre-processing step. As we formally prove in **Theorem 1**, this approach can only capture first-order statistics and will fundamentally fail to detect complex dependencies. For example, consider a triadic relationship where a brain region (C) activates only when its two driving regions (A and B) are in **opposite states** (i.e., one is active and the other is inactive). This triadic interaction can be directly modeled as a **high-order XOR** relationship. However, pairwise statistics would fail to capture it, as **no stable pairwise synchrony** exists between any of the three regions.  Because this initial construction has already discarded crucial high-order information, the entire downstream pipeline is built upon an incomplete representation of the data. \n\nIn contrast, GCM’s ability to learn the graph structure as a whole, guided by the final task objective, empowers it to explicitly search for and encode the vital, high-order relationships. We will clarify this architectural distinction and its theoretical motivation more explicitly in the revised manuscript.\n\n## [W3] Clarification of Table 4 and the role of discretization\n\nThank you for highlighting the lack of clarity in Table 4. To clarify: the column labeled “GCM” represents our full framework with the adaptive **Batch Binarization Algorithm (BBA)**, while the other columns use fixed, static sparsity thresholds.\n\nWe agree that an ablation on a non-discretized variant is an excellent suggestion to isolate the impact of binarization. **We are running this experiment and will incorporate the results into a revised Table 4.** This new comparison will provide a more complete picture of each component's contribution.\n\nFurthermore, we will revise the table caption and improve the paragraph introducing BBA in Section 5 for better contextualization.\n\n## [W4] Inclusion of additional related works\n\nThank you for suggesting these highly relevant references. We agree that discussing them will better contextualize our work:\n\n- Regarding Qiu et al. [1]: This work on learning hypergraphs is indeed closely related, and we are currently **conducting experiments** to provide a direct empirical comparison in the revised manuscript. A key theoretical distinction, however, is that their method requires pre-specifying the number of hyperedges, which introduces a strong prior assumption about the network's high-order structure. In contrast, GCM avoids such direct assumptions. The adjacency matrix learned by GCM reflects the holistic result of high-order co-activation, optimized end-to-end without potential information loss from such structural priors.\n\n- Regarding Kan et al. [2]: While this paper is innovative, its use of a **multi-head** attention mechanism makes the learned structural relationships difficult to interpret explicitly. Thus, it is less suitable for a direct comparison under the umbrella of \"structure learning\" methods. Nonetheless, its success highlights the importance of capturing high-order relationships in brain networks, which aligns with our motivation. We will **add a discussion** on how this attention-based approach relates to our research goals in the revised paper.\n\nCrucially, both of these excellent works focus their analysis on the sample level. A core contribution of our paper is to formally introduce, differentiate, and model FBNs across four semantically distinct resolutions (sample, subject, group, and project). Furthermore, neither of these papers explores the use of top-down global constraints to guide the high-order learning process.\n\nTherefore, we believe includinng them in our paper will further underscore the **originality** and **significance** of our proposed multi-resolution, globally-constrained framework."}, {"review_id": "puHyR1DSIc", "reviewer": "Reviewer_vyUr", "summary": "This paper introduces a new framework for functional brain network modelling named Global Constraints oriented Multi-resolution (GCM), considering not only the typical sample-level modelling, but also including other resolutions from subject, group, and project levels. Experimental results showed significant accuracy improvements with regards to 5 datasets and 2 task settings, while also reporting significant reductions in computational time.", "questions": "As I said, I believe this paper to be really relevant to the community, however, given the weaknesses I have identified, I can only recommend borderline accept, and I will be happy to reconsider my scores if the authors tackle the 4 weaknesses points I've identified.\n\nWith regards to the lack of clarity previously mentioned as Weakness, I left two points here asking for more specific clarifications:\n1. Could the authors clarify some key terms used in the paper, namely what they really mean with:\n\n   1.1. Global constraints. This term is used a lot in Introduction as a key concept of the paper, inclusive in the model's name, but I don't believe this is ever really explained. I would have expected this explained in the Methodology section, but it was not. \n\n   1.2. High-order dependencies/structures/interactions/.... Again, used as a key concept of the paper and in the model's name, but never really defined. I am guessing that this term seems to come from previous literature on hypergraphs specified in the paper, but for such a wide audience like the one in neurips, I'd expect that such a key concept would at least be briefly defined in the main paper such that everyone could understand the main parts of the model. It is even claimed in the abstract that \"Higher-order dependencies\" is theoretically analysed in the paper.\n\n   1.3. FBN structure (learning at 4 levels). The word \"structure\" is used a lot in the paper in the context of learning/optimisation, and from Section 3, we get a more formal definition that an FBN structure is basically an undirected graph, which I understand can be differently learned at each distinct resolution level. However, how is this structure actually learned in the model for the group- and project-levels? In the Methodology section we can see that the label-aligned loss allows for information at sample-level to be learned in the graph, and the subject identity contrast seems to allow for information at the subject-level to be learned in the corresponding graphs, but nothing is said about the remaining 2 resolutions. In my opinion this gets even more confusing at Section 6. At the beginning of it, it is said that the \"FBN structure learned by GCM\" can be used as a \"backbone\" for a GNN classifier. If a structure is a graph, how can it be a backbone, given backbones are typically neural networks? Furthermore, it is not clear what are the \"ground-truth\" group- and project- graphs/labels, to understand what the performance really means here.\n\n   1.4. Inter-subject and intra-subject classification. In Section 6.1 I do not believe this difference is clear at all.\n\n2. Can the authors clarify what are the baselines referred on line 252, with regards to the GCM_{group} model? From table 3, I do not seem to see any corresponding model for direct comparison (only one model for each subject and project levels)", "limitations": "Yes. However, I'd recommend the authors to add a reference for the potential negative societal impacts in the Conclusion section. At the moment only the limitations are pointed to in Conclusion.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "The topic of learning a graph/matrix from EEG or fMRI data is definitely a (still) on-going challenge in the field, and the fact that this paper introduces a way to effectively move away from the typical pairwise Pearson correlations is very significant to the field, with the framework of 4 resolutions being a novel and creative way to frame this problem, as far as I know. The theoretical proof that pairwise-based models might not be enough to recover all the dependencies from the original timeseries is a bit obvious in the sense that functions like the Pearson correlation are not bijective, but having a theoretical proof about this is definitely a strength of this paper. With regards to the quality of the paper, the experiments section is technically sound in the context of the paper, and it's good to see that the authors used quite a varied number of baselines and datasets, while also using more traditional ML models for a fairer comparison. Finally, it is good that the model claims to have more fundamental explainability capabilities embedded in the modelling, which is important to move these discussions towards a more clinical practice, even though the task chosen was the not-so-interesting male/female binary classification.\n\nI am not familiar with the hypergraphs and hyper networks literature, which means I am not completely sure about some of the claims of this paper. However, I identify three main weaknesses on this work:\n1. For the more traditional ML models, I believe that the paper is missing an important inclusion of Random Forests/XGBoost type of models, which I've seen to work better in the neuroscientific contexts when compared to LR and SVM. In this sense, it would be important to indicate what hyperparameters were used here, for instance, what was the kernel type used for SVM?\n2. In an applied neuroscientific paper like this, using only accuracy as an evaluation metric is definitely not enough, and one would expect other metrics like sensitivity, specificity, and AUC ROC, even if left in appendix.\n3. When making a computational analysis like the one reported in figure 3, it is important to have some measure of variability. Furthermore, the paper does not specify if this calculations are related to training or inference.\n4. I do not know whether it's because of my lack of knowledge of the specific hypergraphs literature, but I believe this paper is not clearly written, and I leave some questions about this in the Questions section. \n\n\nSome small points:\n1. I'm guessing \"organismal\" on line 51 is a typo?\n2. On line 161, missing \"is\" between \"It composed\".", "was_revised": true, "final_justification": "The rebuttal provided by the authors was extremely useful for me to better understanding some points and better appreciate the work's contributions. All in all I think this is a very interesting work and I changed my score to Accept.", "rebuttal": "We sincerely thank the you for catching our contribution to the field and the collegial atmosphere you fostered in your review!\n\nYour questions are valuable as they touch upon a core challenge: bridging the objectives of ML and neuroscience. A typical ML paper seeks **prediction accuracy**, while neuroscience uses prediction to learn an **interpretable model** (our FBN). Our work adopts the latter perspective, which we hope clarifies our design choices below.\n\n# Responses to Questions\n\n## Q1: Clarification of Key Terms\n\nThank you for carefully highlighting these core terminologies! Our work intersects **network science, deep learning, and neuroscience**, and our terminology may not have been immediately accessible to all audiences. Your feedback provides a valuable opportunity for us to clarify these concepts, fostering a common ground for this exciting, integrated area of research. We elaborate on each term below and will revise the manuscript to improve this accessibility.\n\n### Q1.1 Global Constraints\n\nThank you for this crucial point! To give a precise definition:\n\n**Global Constraints** are top-down rules that **guide** how we learn a Functional Brain Network (FBN) structure **from the input MTS data**. Their purpose is to make the full adjacency matrix a single, learnable **output** of our system.\n\nIn our work, these constraints **include: 1) an adaptive sparsity based on each FBN's expected edge number, 2) task labels, and 3) subject identity**. We will add this clear definition to the Introduction and detail it in the Methodology (Sec. 5).\n\n### Q1.2 High-Order Dependencies\n\nConventionally, an edge is a **first-order** relationship between two nodes. The simultaneous activation of multiple neurons can be captured as **high-order** dependencies within the network, referring to relational patterns that involve more than two nodes and edges. \n\nIn particular, neural activations that are influenced by combinations of multiple neurons represent another type of high-order dependency. For example, consider a brain region (C) activates only when its two driving regions (A and B) are in **opposite states** (i.e., one is active and the other is inactive). This interaction can be directly modeled as a high-order XOR relationship. However, pairwise statistics would fail to capture it, as no stable pairwise synchrony exists between any of the three regions. \n\nThe crucial distinction lies in _how_ edges are generated. The adjacency matrix of high-order dependencies is learned via global updates, where gradients are applied to the entire structure. This means the resulting edge weights no longer simply encode pairwise (first-order) relationships, but instead reflect interdependent relationships that emerge from the whole-network optimization. We will add this intuitive definition to the Introduction and revise the abstract to better convey \"Pairwise interactions cannot fully capture high-oder dependencies. This limitation is theoretically analysed\".\n\n### Q1.3. Clarifying FBN structure learning at group/project-levels and \"backbone\".\n\nThank you for this detailed question! It highlights several key areas where our explanation was not sufficiently clear. We apologize for the ambiguity and will address each of your points systematically below.\n\n**1. The FBN \"Structure\" as a Learnable Adjacency Matrix**\n\nYou are absolutely correct that the FBN \"structure\" is an undirected graph, represented by its adjacency matrix. The key innovation is that the entire $N×N$ adjacency matrix is treated as a holistic, learnable parameter within our framework, which is optimized end-to-end.\n\n**2. How Group- and Project-Level Structures are Learned**\n\nWe sincerely apologize for not making this mechanism explicit in the paper. The learning process for all four resolutions relies on the same loss function, but the difference lies in how this learnable adjacency matrix is shared across samples:\n\n- **For Sample-Level**: Each sample $\\mathbf{X}_i$ has its own unique FBN adjacency matrix $\\mathbf{A}_i$, which is updated only by the loss computed on that sample.\n\n- **For Subject-Level**: All samples from the same subject share one single FBN adjacency matrix $\\mathbf{A}_{subject}$. Gradients from all samples belonging to that subject are aggregated to update this shared matrix.\n\n- **For Group-Level**: All samples belonging to the same group (e.g., a specific clinical cohort or, in our case, a task label like 'female') share one single, common FBN adjacency matrix. For instance, when processing samples with the label 'female', the gradients on these samples are used to update the single, shared 'female' group-level graph.\n\n- **For Project-Level**: Similarly, all samples in the entire dataset share one single project-level adjacency matrix. Gradients from all samples, across all batches, contribute to the update of this one global graph.\n\nEssentially, for group and project levels, the adjacency matrix acts as a shared, learnable parameter that accumulates structural information from all relevant samples. We will explicitly add this detailed explanation to Section 5 of the revised manuscript.\n\n**3. \"Backbone\" Terminology**\n\nThe term \"Backbone\" refers to using the learned FBN structure as the basis for massage passing on GNN. We agree that \"backnone\" is imprecise in this context. \"Computing graph\" or \"input structure\" for the GNN classifier is indeed more accurate. We will correct this terminology.\n\n**4. \"Ground-Truth\" and Evaluation at Higher Levels**\n\nThank you for raising this crucial point. Across all four resolutions, the prediction task itself is identical: predicting the label of each sample.\n\nThe key difference, which addresses your question about \"ground-truth\" for higher levels, is the role of this label, or information. For the group-level, the label (e.g., 'female') serves as a mapping mechanism: it directs the framework to associate all samples with the 'female' label to the single, shared 'female' group-level FBN. The gradients from predicting the sample-level labels of all these individual 'female' samples are then aggregated to update this one representative graph.\n\nTherefore, the accuracy score here is an index of the learned FBN's **quality and representativeness**.\n\nThis reframing gets to the heart of our paper's contribution. We leverage a standard ML task (classification) not as an end in itself, but as the supervisory signal to achieve a neuroscience goal: learning an interpretable, high-order brain network. We will make this distinction much clearer in the revised manuscript.\n\nWe hope this systematic breakdown fully clarifies the learning mechanism at all resolutions. Your questions have helped us identify a significant gap in our explanation, and the revised manuscript will be much stronger for it.\n\n### Q1.4 Inter- vs Intra-subject Classification\n\nWe apologize for the lack of clarity. We will revise Section 6.1 to explicitly state:\n\n- **Intra-subject classification:** Training and testing sets are created _for each subject individually_. This assesses if a model can learn stable, subject-specific patterns over time.\n    \n- **Inter-subject classification:** Subjects are split into training and testing groups. This assesses if a model can learn patterns that generalize across different individuals.\n\n## Q.2 Clarify the baselines and evaluation of GCM$_\\text{group}$\n\nThank you for this sharp observation! You are correct; direct baselines for GCM$_\\text{group}$ are absent due to a fundamental paradigm difference. \n\nFrom a conventional machine learning perspective, our group-level setup could raise concerns about \"label leakage,\" which is precisely why prior methods have typically avoided this modeling approach.\n\nHowever, as we clarified in our response to Q1.3, our primary goal is not prediction for its own sake, but to learn a generalizable structural representation for a cohort. Learning such cohort-level FBNs is an essential and often unavoidable task in computational neuroscience (e.g., for identifying biomarkers for a clinical group).\n\nTherefore, we deliberately isolated this resolution to propose a valid framework for this important, domain-specific problem. This novelty is why direct competitors are not readily available, a point we will clarify in the revision.\n\n# Addressing Listed Weaknesses\n\nWe also appreciate the your detailed summary of specific weaknesses. Below, we respond to each of these concerns in turn.\n\n## W1. Lack of traditional baselines\n\nThank you for your insightful advice on strengthening our baseline comparisons. You are correct that Random Forest and XGBoost are powerful classifiers in this domain, and we agree their inclusion will make our empirical evaluation more comprehensive. We will add them to the revised paper and clarify that the SVM used an RBF kernel.\n\n## W2. On the use of additional evaluation metrics beyond accuracy\n\nThis is an excellent suggestion! We will add tables reporting sensitivity, specificity, and AUC-ROC for all relevant experiments in the appendix.\n\n## W3. Clarifying variability and training/inference cost in Figure 3\n\nThank you for your detailed review! Figure 3 reports the average **training time**. We will revise the figure to include error bars representing the standard deviation to show variability.\n\nRegarding inference cost, we will clarify in the revision that for GNN-based models like ours, inference is extremely fast (e.g., **on the order of milliseconds per sample**), making it a non-factor for practical application compared to the training cost.\n\n## Response to seggestion on limitation and societal impacts\n\nThank you for this thoughtful suggestion! In the revised version, we will add a sentence to the Conclusion section that points to the potential negative societal impacts in Appendix K.\n\n## Typos and minor issues\n\nThank you for your careful reading! We will thoroughly proofread the manuscript to eliminate any remaining errors in the revised version."}]

Author rebuttal:

Meta-review: This paper introduces a Global Constraints-oriented Multi-resolution (GCM) framework for learning functional brain networks. The main strength of the method lies in its multi-resolution hierarchies for capturing interactions at different scales and its demonstrated effectiveness over previous methods, receiving three positive reviews and one negative review. Reviewer eH9B raises concerns about 1) the definition of “global constraints,” 2) the significance of the ablation study, and 3) particularly the differences from some missing baselines. During the author-reviewer discussion, the authors provided detailed explanations of the motivation and additional results, which adequately addressed the first two weaknesses. For the third weakness, the authors acknowledged the similarity between the baselines, while providing sufficient intuitive and experimental comparisons with the mentioned baselines, such as BNT to show the theoretical and empirical differences. The AC believes that, with the inclusion of the detailed supporting information from the rebuttal, this paper makes a meaningful contribution to functional brain structure learning from both theoretical and experimental perspectives.
Decision: Accept (poster)