Where Graph Meets Heterogeneity: Multi-View Collaborative Graph Experts
Authors: Zhihao Wu
Jinyu Cai
Yunhe Zhang
Jielong Lu
Zhaoliang Chen
Shuman Zhuang
Haishuai Wang
Venue: NeurIPS 2025 poster ()
Keywords: Graph neural networks
heterogeneous data
multi-view learning
Primary area: deep_learning

Abstract: The convergence of graph learning and multi-view learning has propelled the emergence of multi-view graph neural networks (MGNNs), offering strong capabilities to address complex real-world data characterized by heterogeneous yet interconnected information. 
While existing MGNNs exploit the potential of multi-view graphs, the inherent conflict persists between the two critical inductive biases of multi-view learning, consistency and complementarity. Consequently, the challenge of defining and resolving this tension in the new context of multi-view graphs remains largely underexplored. To bridge this gap, we propose Multi-view Collaborative Graph Experts (MvCGE), a novel framework grounded in the Mixture-of-Experts (MoE) paradigm. MvCGE establishes architectural consistency through shared parameters while preserving complementarity via layer-wise collaborative graph experts, which are dynamically activated by a graph-aware routing mechanism that adapts to the structural nuances of each view. This dual-level design is further reinforced by two novel components: a load equilibrium loss to prevent expert collapse and ensure balanced specialization, and a graph discrepancy loss based on distributional divergence to enhance inter-view complementarity. Extensive experiments on diverse datasets demonstrate MvCGE’s superiority.

Where Graph Meets Heterogeneity: Multi-Vier Collaborative Graph Experts
Zhihao Wu's Jinyu Caf Yunhe Zhang', Jielong Lu', Zhaoliang Chen', Shuman Zhuang", Haishuai Wang't
ZZejjang Key Laboratory of Accessible Perception and Intelligent Systems, College of Computer Science and Technology, Zhejiang University
Institute of Data Science, National University of Singapore Department of Computer and Information Science, University of Macau
'Department of Computer Science, Hong Kong Baptist University CCollege of Computer and Data Science, Fuzhou University
(zhhihaowu1999,jaa ityyuuua9999 1995, changyhannie)@gue chenz1236outlook. com, shumanzhuang0163. com, haishuai. wwwgai/..nngee....oooo edu. cn
com, jielonglu2022016. com,
Abstract
The convergence of graph learning and multi-view learning has propelled the emer- gence of multi-view graph neural networks (MGNNs), offering strong capabilities address complex real-world data characterized by heterogeneous and intercon- nected information. While existing MGNNs exploit the potential of multi-view graphs, the inherent conflict persists between the two critical inductive biases of multi-view learning, consistency and complementarity. Consequently, the chal- lenge of defining and resolving this tension the new context of multi-view graphs remains largely underexplored. To bridge this gap, we propose Multi- view Collaborative Graph Experts (MvCGE), novel framework grounded in the Mixture-of-Expeee (MoE) paradigm. MvCGE establishes architectural consistency through shared parameters while preserving complementarity via layer-wise collab- orative graph experts, which are dynamically activated by graph-aware routing mechanism that adapts to the structural nuances of each view. This dual-level design is further reinforced by two novel components: load equilibrium loss prevent expert collapse and ensure balanced specialization, and graph discrepancy loss based distributional divergence to enhance inter-view complementarity. Extensive experiments on diverse datasets demonstrate MvCGE's superiority.
Introduction
With the advancement data collection and analysis capabilities, modern data no longer treated isolated singular. Instead, entities now understood to be deeply interconnected through complex and pervasive relationships, which are captured with increasing fidelity. Graphs, due to their universality, widely used represent these relationships at capturing complex dependencies and have become cornerstone of machine learning, leading to advanced techniques for learning representations from structured data (Liang et al.) 2024 Cai el al.) 20244 Dai el al. 2016). Beyond inter-entity relationships, the 'non-isolation' of data is also manifested in the heterogeneous information sources, referred different views. For instance, in social networks, user may be associated with various types of interactions (e.g., friendship ties behavioral interactions); With multimedia systems, objects be described using multiple
powerful modeling tool. They excel
"Equal contribution.
Corresponding author: Haishuai Wang.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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modalities (e.g., visual features and semantic tags). This diversity introduces unique challenges. giving rise to new paradigms like multi-view learning [Liang et al. 2025] Zhang er al. 2025 Chen er
al. 2023a Xu et al. 2015]. For such heterogeneous data, prior work in multi-view learning provides a foundational framework built upon two well-known key properties. Specifically, each view provides unique information that complements the others, which is the property known as complementarity.
Since these views are rooted in the same underlying entities, there is an inherent agreement between them, referred to as consistency. These two distinct characteristics enable multi-view learning to outperform the single-view counterpart (Tao et al. 2019]. But these characteristics may inherently conflict [Dong et al. 2023]. Overemphasis on either distinctive or shared information will diminish the advantage of multi mueti-view learning. Therefore, the effective promotion of a balance between these two properties has become key focus.
Both the universal inter-relationships and increasing views further extend the heterogeneity of data from different perspectives. But interestingly, graph learning uses topology as unifying structure, providing a common ground to anchor, align, and integrate diverse information. Researchers have found that graph-based multi-view learning demonstrates promising performance (Li et al. [0255] Wan et al. 2024]. For example, ZZhng et al.) [Cen el al. facilitated view alignment by constructing graphs, which significantly enhanced clustering; while Duan et al.) Huang et al.) focused on semi- supervised learning, fully exploiting the graph to amplify supervisory signals. However, these methods remain traditional, lacking deeper understanding of graph theory, and mainly build on the concept of similarity preservation, struggling to cope with complex data. There is widespread interest in empowering multi-view learning with advanced graph techniques in the era of deep learning.
Recently, Graph Neural Networks (GNNs) have rapidly developed and become a powerful technol- ogy for graph learning. Spectral GNNs [Deffer- rard et al. 20166 effectively extend convolutions to non-Euclidean graph data; Graph Convolutional
GhNN Layerr
GANN Lagw
GNNi Layar
GHN Layer
ONN Layer
Network (GCN) (Kipf and Welling 2017]. estab- lished upon this, was then proposed and quickly became a phenomenal model. Due to the power- ful capabilities of GNNs [Xu et al.) [019 and the synergistic effect between multi-view learning and graphs, Multi-view GNNs (MGNNs) have become an effective paradigm. Distinct from prior works, MGNNS typically explicitly formulate heteroge- neous data as multi-view graphs and utilize modern GNN architectures. Although the graph structure is a powerful instrument for harmonizing heterogene- ity, the classic dilemma between consistency and complementarity persists. The primary challenge shifts to how to define and resolve this long-standing conflict within the new context of multi-view graphs. Existing methods have made attempts, specifically, many MGNNs adopt unified model that naturally allows for high collaboration between views through shared weights [Lin et al. 20244 Chen et al.) 2023b) [heng et af.) [2020) Xia et al.) [022]. But this may lead to an overemphasis on consistency, thereby undermining complementarity. Also, in graph signal processing. graph filter may fail to match the potentially heterogeneous multi-view graph structures. Another pipeline builds a multi-channel model [Chen et al. 2024a; [u el al. 2024a Wu el al.) 2023 Li el al. 2020). where unique parameters are assigned to each view, ensuring complementarity. Because the interaction be- tween views relies solely on post-fusion, this approach may sacrifice intrinsic inter-view interactions and lack architectural flexibility. The two existing pipelines are depicted in Figure [ This raises critical question: How can we design a reasonable MGNN paradigm that achieves a balance between consistency and complementarity?
Viow
Gruphi Excert
Groph Expert
Gruph Export
Graphi Excert
Graph Expert
Graph Expert
Fite
View
Vew V
Figure I: The illustration of MGNN paradigms based on (a) shared model, (b) multi-channel model, and (c) MvCGE
In response to this challenge, we introduce a flexible and effective MGNN framework, termed Multi-view Collaborative Graph Experts (MvCGE) in this paper. Figure 2 briefly illustrates the architecture of MyCGE. First, by reviewing existing MGNNs through the lens of graph spectral theory, we identify a critical limitation: shared graph filters cannot effectively handle the heterogeneous structural properties of multi-view graphs. Drawing inspiration from Mixture of Experts (MoE), we propose layer-war wise collaborative graph experts as the foundational building block of MvCGE. This block enables distinct graph filters, or regarded as GNN layers, to collaboratively process multi-view
2

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graphs, thus preserving view-specific information. Notably, the experts are shared across views, establishing an architectural foundation for consisteney. Moreover, the number of graph experts is decoupled from the number of views, offering better flexibility. The incorporated graph-aware routing allows for the dynamic selection of suitable graph experts based on the structural characteristics of each view. To safeguard against model collapse and the consequent loss of complementary information, we introduce two novel mechanisms. The load equilibrium loss ensures a balanced allocation of training effort among the experts, avoiding dominance or over-proliferation of certain experts. The graph discrepancy loss, grounded in graph isomorphism, treats multi-view graph latent representations discrete distributions and encourages experts to learn distinct representations, preventing collapse into the same latent space. The main contributions of this paper are:
We thoroughly review existing MGNN pipelines and reveal critical challenge, i.e., shared or view-specific models fail to flexibly adapt to multí-view graphs.
We propose MvCGE, a novel MoE-inspired MGNN framework, effectively balancing requirements for consistency and complementarity in handling heterogeneous data.
Two innovative mechanisms are introduced address model collapse and enhance the protection of view-specific information from different aspects.
Experiments on both single- and multi-view graph datasets demonstrate that MvCGE out- performs competitors, showcasing impressive performance and generalization capabilities.
Related Work
Multi-view Learning Multi-view learning [Luu et al. 2023 2022
Chen et al. 2022a leverages
information from multiple perspectives to enhance performance, with consistency capturing shared information and complementarity harnessing unique contributions. Graph-based approaches, known for modeling arbitrarily distributed data, have gained prominence in this field (Sun et al.) 2024 Wan et al.) [023 [i el al. (209). Nie ef al. proposed to automatically weight view-wise graphs in multi-view clustering. Li et al. utilized graphs to enhance incomplete multi- learning. [Wen er al. leveraged graphs to refine cross-view correspondences. However, these methods fail to fully exploit graphs' potential and lack integration with deep learning advancements.
Mixture of Experts The concept of MoE be traced back earlier works [aacobs al. 1991
Jordan and Jacobs) 1994), designed for dynamically combining multiple specialized models to solve complex tasks. Modern advancements have adapted MoE deep learning, making powerful in vision and language models [Shazeer et al. [0177] Lepikhin et al., 2021 Fedus et al. 2022; Fang el al. 2022 Shen al. 2025 Miao el al.) 2022], However, the potential of MoE in multi- view learning remains unexplored. Meanwhile, graph MoE is still in its infancy, with only a few researchers
attempting to develop MoE for graph-baser based tasks (Ma er al. 2024, Wang al. 2023 Hu al.
2022]. There remains a lack of investigation for multi-viww graphs [Wan et al. 2023].
Graph Neural Networks GNNs are a crucial tool for graph-based machine learning CCai et al.
2025 Wang al. 2025 Fang el al. 2025b al and can be broadly divided into spectral and spatial approaches. Spectral GNNs, utilizing spectral graph theory to extend convolution operations to non-Euclidean space, once dominated the field. Kipf and Welling intrrduuced the GCN, bridging the gap between spectral and spatial paradigms, and laying the foundation for modern GNNs. Building on these successes, MGNNs have emerged to tackle challenges multi-view learning. Current MGNNs follow two main pipelines. One strategy constructs a unified model with shared parameters to process all views jointly, inherently ensuring consistency across views. To achieve this goal, Chen er al. proposed fusing multi-view graphs, while Lu et al, developed graph generation approach. The other strategy employs multi-channel architectures, learning separate parameters for each view, as exemplified by [Lin ef al. 2024 Chen et al.) 2024b). However, these two pipelines often lean toward two extremes: overemphasizing either consistency or complementarity. The former sacrifices view-specific information, while the latter may cause divergences and misalignments between views.
3

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Load Equiibrium Loss Zogm
Graph Distrbution Discropancy Loss Leup
a
Functions
Lyzz
Graph Expert f af
Graph
Ouput
Graph Expert ag
Graph Readout
Class
Rig
Ag
Graph
Ourput
Graph-seave Routing
Mull-siew Aggragation
Graph Export fe
Topst Gaagi
Ch
Gmap-aware Routing
Graph Sy
Collaborative Graph Experts
Exper: Output
Oulput
Multi-view Graph Input
Layer-wise Architecture
View-apecific Output Aggregation
Multi-view Collaborative Graph Experts
Tasks
Figure 2: The illustration of a single layer of the proposed MvCGE.
3 Methodology
Notation We define attributed graph as EV, X), where the node set with cardinality [V] N, and & is the edge set. The graph's topology is represented by its adjacency matrix A {0, N The node attributes are represented by a feature matrix X R RN where D is the dimension of the node features. The degree matrix D E N is a diagonal matrix where D.: = E The graph Laplacian is defined L D A or L -- D à AD for its symmetric version. A multi- graph describes set of graphs {9e associated with heterogeneous information. The v-th graph can be defined as G, = (V, E X) with corresponding adjacency matrix A, and shared attribute X.
[V] where the node set
this subsection, we will discuss existing paradigms from the lens of graph filters. To begin with, it is known that the Laplacian matrix be decomposed L UAUT, which essentially induces a Fourier transform on the graph domain, where eigenvectors correspond to Fourier components and eigenvalues represent frequencies of the graph. Take X as the input signal, the general form of a graph convolutional layer can be written as
Motivation
= ff,
where fe is function parameterized with O. For the early spectral GNNs [Bruna al. 2014 the function is specified as he (L),, where hL(L) [ is a parameterized graph filter. Due to the high compu- tational burden, researchers then proposed approximate the graph filter by polynomial [Defferrard
et al. 2016), as
fe X)
O,L'X,
where + {0, :t [T]} are the parameters. Defferrard er al. 2016 achieved fast localized filters by using Chebyshev expansion, and the famous GCN [Kipf and Welling 2017 further fixed T = 2 and simplified the convolution bridge the spectral and spatial GNNs a unified form. Therefore, without loss of generality, we describe the recursive formula of multi-layer modern GNNs:
(3) where O E Ni is the input of the 1-th layer, which is initialized by H X, and ( can be regarded as the extended multi-channel parameters of the graph filter. Training GNNs through optimizing the graph filter parameters ( in each layer results in their powerful performance in graph-based machine learning tasks.
H (1+1) = fem (G, ())).
These GNNs have been rapidly applied to more complex and realistic multi-view graph scenarios. To ensure consistency across views, many approaches typically adopt a unified and shared model across all views. This strategy naturally enables highly collaborative learning among views by the weight

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sharing, which leads to good consistency, but it often struggles to effectively model the multiplex graph structures inherent to diverse views. Considering any two views, we typically aim for the graph filter to produce results that exhibit consensus rather than significant disagreement. However, the effectiveness of a shared graph filter applied two graph structures is significantly influenced by the spectral gap between these graphs. Discussions on the transferability or stability of graph filters are usually based on demanding assumptions [Ruiz et al.) 2020, Gama et al. 2020; Levie et al. [021]. In other words, when the structures of the two graphs differ substantially. it becomes challenging to ensure that the graph filter achieves optimal performance across both views. An intuitive solution is to assign separate graph filter to each view, allowing each view to learn parameters tailored to its own needs. This approach has been widely considered. However, while it takes into account more complementarities compared to shared filter, it inevitably swings to the other extreme by sacrificing consistency in the architecture. Furthermore, the number of sub-modules is directly related to the number of views, which limits its flexibility. To address these limitations, we aim to engineer an MGNN framework that ensures scalable integration of inter-view dynamics, thereby better capturing and balancing the consistency and complementarity.
Overall Framework To begin with, let us consider the input from the v-th view and define the following layer-wise formulation:
[(+1) Fac (Go, H!)),
where H 4 denotes the v-th graph embedding at the 1-th layer with H 0 = X, and the function Fao:
with trainable parameter set (0( is assembled by multiple graph convolutions:
Fac0 (Gu, H = Ro (ffow) (Gu. H(!) m E [M]}},
(5)
where Ryt denotes the routing function parameterized by (1(), and a (1 e :m [M]] Building upon this, we introduce an ensemble of multiple graph filters, For., aiming to leverage graph filters equipped with distinct parameters to collaboratively process graph information from different views. Specifically, by designing the router Ryper, appropriate combinations of graph filters are selected to handle potentially heterogeneous structures from various views. Subsequently, we consider processing multi-view graph and constructing a multi-layer network. The proposed MvCGE framework calculates the final representation H NxC by
H = A= (H = Fa (9.. X) E [V]]),
(6)
where H, E R is the w-th learned final embedding, A= is multi-view aggregation function with parameter set E, and Fg is defined by concatenating L layer-wise functions:
F == Fo
o Fa(t) o... Fou-
The entire function FRcn is expected to be shared across views, thereby facilitating inherent inter-view interactions. In our framework, F4 is shared across views, and this characteristic ensures sufficient interaction between the graphs of different views at the architectural level. That is, all parameters & in the model are trained across the graphs of all views capture consensus. Additionally, each layer F&.0 of F& integrates multiple graph experts fo which ensures that the model has sufficient capacity to adapt to the data of each view, thereby capturing their complementarities. Note that the number of graph experts M is independent of the number of views V. which provides the model with additional flexibility. Therefore, MvCGE strikes a balance between the inherent requirements for consistency and complementarity in multi-view learning. After outlining the framework of MvCGE, the following subsection will discuss the details and specific implementation.
Layer-wise Collaborative Graph Experts For the aforementioned design ideas, we implement our framework by drawing inspiration from the concept of MoE. Formally, given the input of the e-th view in the 1-th layer !!. we specify that Fate aggregate decisions from several graph experts by
Σ
I.. fo (Gv. H H!!)
(8)
mE[M]
where roo E NMM the w-th routing matrix that contains the routing weights of all the graph filters in the 1-th layer, which acts as graph experts in this framework. Through this layer-wise design,
5

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each layer of MvCGE can be viewed as an independent MoE and is controlled by a specific router, allowing for more flexible selection and combination of graph experts.
Next, we specify the routing function Rgt,, whose role is to select the appropriate combinations of experts for each node in each view. We propose improved graph-aware top-k gating mechanism. Specifically, we denote the gating weight of the i-th sample in the v-th view to the m-th expert as
J Softmax (ggge (G.,, H()) )c:)
s.t. || = K, Vi E [N],
where the gating module Iw(0) outputs M -ddimensional vector acting as the weights for integrating outputs from diverse experts. Note that this module has a constraint to ensure sparse gating. In multi-view graph learning scenarios, the design of gom is critical, as is key to preserving view specificity, It must both perceive the graph and view-level patterns.
groe (Go, " ) = Topk (IP(I) L... E (.)))P(( gole.
where P is pooling function and S, is the node set associated with a sampled local subgraph of node Therefore, P ((((H))) J... E S.}) and P(H respectively encode the local graph structure and view-specific global information. W gate E 2 is the trainable parameter matrix mapping the input to gating scores, respectively. We adopt Top-K sparse gating: for each node i, only the Top-K K expert logits are kept while the rest are masked out (set to -c) before applying softmax. This yields [|[ Jt.: llo = K with the remaining weights being exactly zero, such that only the experts with top- K gating weight are selected.
When processing large-scale graphs, graph convolution operations can become major bottleneck due to the massive number of edges. The employment of multiple graph experts imposes additional computational overhead, Therefore, we consider further specializing the graph experts and propose an accelcration trick. We formalize the graph expert as
Acceleration Trick
(g H) = ((P.H.) OO!!)
(10)
where ( is a non-linear activation function, P. is a filtering operator for the view v, and @! E D is the trainable weight matrix. Here, we consider class of simplified spectral GNNs where the filter parameters are treated as being encoded in the trainable weights @!!) thereby decoupling a parameter-free operator P, For example, it can be the Chebyshev polynomial >" T, (I.,) for ChebNet [Dffferrard al. 22016). and I L, for GCN (Kipf and Welling 2017]. For this class of graph experts, we only need to compute the parameter-free graph filtering once per MvCGE layer. This result is then shared among different experts, which apply their distinct trainable weights for different graph convolutions. This considerably accelerates the entire framework.
Regularization Although MyCGE ensures consistency through a flexible architecture, its training faces challenges that could lead to model collapse, thereby losing the complementary information between views.
Load Equilibrium: From the optimization perspective, due to the gating mechanism (hhazeer et al. 2017]. in some cases MvCGE may overly rely on group of graph experts while preventing others from being trained. This imbalance is selffreinforcing, causing the selected experts to proliferate, while other experts become increasingly unlikely to be chosen. Ultimately, MvCGE will collapse into a model with only a few experts activated, losing the advantage of capturing view-specific information. To address this, we introduce load equilibrium loss. For the w-th view, let
P EE kim WH
Σ II(T)).. > 0}
be the average gating weight and load of expert m, we have
1 V
M Σ
Logm
w
P
log
(11)
V
M
1=1
where we consider the intra-view and then the inter-view equilibrium. By minimizing Corms each graph experts are encouraged to have similar selection scores and frequency.

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Graph Diserepancy: From the perspective of the latent space, even if load equilibrium is ensured and different graph experts are given equal training opportunities, they may collapse into similar latent spaces. Different graph experts are supposed to perform distinct filtering on multi-view graphs, rather than collapsing into a single graph filter. To address this challenge, we take single-layer MvCGE as an example to elucidate the computational mechanism of the graph discrepancy loss. Given the v-th view graph G., let fe... and fe... denote any two distinct graph experts, the expert-specific embeddings are defined as:
Z [Tl.. (9:
(12)
where r, denotes the routing matrix and z denotes the expert-specific embedding generated by expert fe... Similarly, we obtain Z for the other graph expert. For clarity, we omit the layer index in this single-layer illustration. Then the two embeddings are fed into the readout function, or called pooling function as
2 = P-
(13)
Here, -..h can be interpreted as a global graph embedding that encapsulates the semantic and structural information captured by expert fe... Recent studies have demonstrated that readout functions paired with GNNs can effectively perform graph isomorphism computations [Xu et al., 2019]. Therefore, we combine the readout function with layer of GNN to form model Q. Then for all V views, these graph embeddings {z, V E [V] are treated as samples drawn from a distribution Io e induced by
the function fe... where Q.. P fee-
After that, leverage Maximum Mean Discrepancy (MMD) (Gretton et al. 2012 to quantify the discrepancy between distributions from different graph experts:
2E P |m(z"" z }], Mi (Pe) Iue.. me,.. 1l Be [<(z"" z.) Be [r(2""' 2.
(14)
where & is a kernel function, e.g., Radial Basis Function kerne k(z, z') = exp( |K==z' 12 B). Graph 20
embeddings z and z' are sampled from distribution Po and oor respectively. Nro and Moz are the means of distributions Pe.... and Pe.., in the reproducing kernel Hilbert space (RKHS) H. respectively. Then we introduce the generalization MMD
(PP
sup {llme me lBx : EEE
(15)
where IC is family of kernels. According to (Gretton et al. 2012]. we specifically estimate MMD by
Ma (P
sup KER
V3
1
x(2
+ 1
1
x 2
2 V2
ΣΣ
m x(z
2.
(16)
Consequently, considering the multi multi-layer case, we propose the graph discrepancy loss:
Ldsp - Σ M(M - 1) mcmm
Σ
My
o0 P (00) PO.)
(17)
where 0.6l = Po Se is a model combining a readout function and a graph expert. Utilizing the two losses together ensures that the graph experts receive similar opportunities and learn meaningful latent representations, thereby promoting MyCGE to capture complementary information.
Training Strategy
The proposed MvCGE is trained using the following objective function
Liotal Lisk + aLegm + 3Lasp:
(18)
where Lisk is the downstream task loss. The widely used Cross-Entropy loss is adopted for the semi-supervised classification task:
Lrsk
-- Yig log Yi...
(19)
VERI
where Y, Y E Rlixe are predicted and groud-truth labels, and Ω is the training set. Note that Y = Softmax(H,.)
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Table 1: Macro Fl and Micro Fl scores of all methods on multi-view datasets (mean value of ten runs), where the best and the second-best results are highlighted in orange and blue, respectively.
Datasets Metries
ACM-M
DBLP
IMDB
YELP
AMINER
MaF1
MiFI
MaF1
MiFI
MaFi
MiFI
MaFl
MiFI
MaF1
MiFi
GCN
0.7860
0.7881
0,9012
0.9163
0.2429
0.5544
0.5196
0.6743
0.6842
0.8156
DGI
0.2235
0.3690
0.2432
0.3763
0.2626
0,5519
0.5034
0.6831
0.3449
0.6460
HANN
0,9147
0.9139
0.8927
0.9039
0.2389
0,5586
0,4829
0.4893
0.7234
0.8476
DMGI
0,8666
0.8681
0.6567
0.7106
0.3533
0.5726
0.5161
0.6985
0.3029
0.6546
IGNN
0,8290
0.8270
0,8681
0,8750
0.4531
0,5481
0.6449
0,7123
0.7453
0.8521
MRGCN
0.8758
0.8745
0,8949
0.9047
0.4517
0,4769
0.5435
0.7370
0.7335
0.8294
SSDCM
0.8765
0.8763
0.8942
0.8990
0.4940
0.5910
0.5270
0.7020
0.2620
0.5548
MHGCN
0.8887
0.8907
0,9300
0,9360
0,5154
0,6423
0.6085
0.7328
0.7500
0.8517
AMOGCN
0.9241
0.9238
0.9227
0,9280
0,5016
0.6506
0.6753
0.7241
0.7570
0.8338
JIMGE
0,9080
0.9066
0,9156
0.9235
0.3293
0.5735
0.5857
0.7484
0.7371
0.8478
MvCGE
0.9348
0.9245
0,9256
0.9317
0.5157
0.6883
0.7776
0,8008
0.7897
0.8640
Table 2: Macro Fl and Mícro Fl scores of all methods single-view datasets (mean value of ten runs), where the best and the second-best results are highlighted in orange and blue, respectively.
Datasets
ACM
Citescor
CoraFull
Flickr
UAI
Metrics
MaF1
MiFl
MaF1
MiF1
MaF1
MiFI
MaF1
MiFi
MaF1
MiF1
GCN
0.8827
0.8842
0.6523
0.6909
0.5277
0.6279
0.5002
0.5104
0.4506
0.5851
SGC
0.8116
0.8087
0.6157
0.6642
0.5320
0.6292
0.4422
0.5101
0.4679
0.5652
APPNP
0.8837
0.8824
0.6510
0.6972
0.5236
0.6395
0,5124
0,5212
0.4655
0.6189
JKNet
0.8524
0.8545
0.6813
0.7292
0,5126
0.6268
0,5112
0.5431
0.4134
0,5623
DAGNN
0,8699
0.8917
0.6812
0.7290
0.5353
0.6481
0.6214
0.6065
0.4778
0.5927
GCNII
0,8978
0,8980
0,6712
0.7064
0,5828
0,6403
0,5904
0,5771
0.4869
0,6382
GNNHF
0,9052
0,9067
0.6760
0.7249
0,5846
0.6672
0.5867
0.6078
0.4495
0.5764
AMGNN
0.8996
0.8993
0.6873
0.7292
0,5664
0.6185
0.7580
0.7569
0.4787
0.6431
HiDNet
0.8958
0.8961
0.6661
0.7146
0,5304
0,6531
0.6200
0.6333
0.4568
0.6165
AGNN
0.8973
0.8974
0.6264
0.6721
0.5065
0.5930
0.5721
0.5844
0.4697
0.6206
MvCGE
0.9110
0.9114
0.6908
0.7356
0.5924
0.6693
0.7675
0.7790
0.4797
0.6591
4 Experiments
Experimental Setting To evaluate the effectiveness of MvCGE, we conducted experiments on ten datasets, including five multi-view graph datasets (ACM, DBLP, IMDB, YELP, AMINER) and five typical single-view graph datasets (ACM, Citeseer, CoraFull, Flickr, UAI) plus one large-scale multi-view graph Freebase and one large-scale single-view graph OGBN-arXiv. To adapt MvCGE to single-vies graphs, we generate a supplementary view for each single-view dataset based on the k-Nearest Neighbor algorithm, where we set k 10. Note that, to distinguish between the different ACM datasets, we denote the natively multi-view graph ACM as ACM-M. For different types of datasets, we compared MvCGE with various competitors, including nine methods designed for multi-view datasets (DGI /Velickovic el al. | 2019), HAN (Wang et al. [209)], DMGI (Park el al.
2020). IGNN [Gu et al. [2020) MRGCN [Huang et al. 2020). SSDCM [Mitra et al.) [021]. MHGCN Yuu et al.) 2022]. AMOGCN [Chen et al. 20224b. HMGE [Abdoss et al. | (024] and ten methods designed for single-view datasets (GCN [Kipf and Welling
2017]. SGC [u et al. [019]. APPNP
[Gasteiger et al. 2018), JKNet [Xu et al. [018), DAGNN
[Li et al. 2020 GCNII [Chen et al.
2020). GNNHF [huu al. 2021]. AMGNN (Zhu et al.
2023 HiDNer et [Li et al. 2024
AGNN
Chen et al. 2023cl).
In this subsection, we conduct comprehensive evaluations of MvCGE's performance in semi-superviset classification tasks across diverse datasets, empirically demonstrating its superior performance compared to stafe-of-the-art methods. Experimental setups and analyses on these datasets are as follows: 1) Multi-view Graphs. Here, the training ratio is set to 20% and the Macro Fl and Micro Fl scores recorded in Table I The experimental results reveal that the proposed
Performance
8

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MHGCN
AMOGON
HMGE
McGGE
Figure 3: The visualization of representations learned by MHGCN, AMOGCN, HMGE, and MvCGE on the YELP dataset.
GCN
DAGNN
GNNHH
MvOGE
Figure 4: The visualization of representations learned by GCN, DAGNN, GNNHE, and MvCGE on the ACM dataset.
MvCGE outperforms baseline methods across most datasets. Notably, substantial performance gains are achieved on the Yelp dataset, primarily due to the inherent limitations of meta-path- bath-b approaches in effectively modeling complex non-linear cross-view relationships. In contrast, guided by the principles MoE, MvCGE effectively resolves the consistency-compleeee complementarity trade-off while preserving view-specific information in multi-view learning, as validated in our experiments. 2) Single-view Graphs. Following the commonly used semi-supervised node classification settings, we randomly select 20 samples per class for training. 500 samples for validation, and 1,000 samples for testing, with the detailed results presented in Table 2 The figure shows that MvCGE consistently outperforms all baseline methods and exhibits strong compatibility and adaptability to single-view datasets. Additionally, as depicted in Appendix, we also test MvCGE on the large-scale graph
OGBN-arXiv and Freebase, showing its scalability.
Visualization We employ t-SNE Van der Maaten L 2008 to visualize the learned representations of top-performing baselines on the YELP (Figures 3) and ACM datasets (Figures D. Each node is represented point, color-coded by its class label, with contour lines indicating category density distributions. Complementary histograms along the axes reveal the embedding distributions across dimensions. The visualizations demonstrate that MvCGE achieves superior cluster separation with well-defined class boundaries and compact intra-class distributions. Analysis the histogram distributions reveals that MvCGE achieves better-separated representations in the feature space.
Sensitivity In this section, we assess the impact of varying the parameters a and 3 on the perfor- mance of MvCGE, where a and balance the loss Ceqm and Lasp. as depicted in Figure (a) and (b). It can be seen that model performance exhibits similar stable fluctuation patterns for both a and across ACM and YELP datasets, with the optimal performance achieved at a le- le and 3 = 1 or - demonstrating that the integrated two losses effectively capture complementary information and consistently enhance performance when appropriate parameter values are selected. Beyond this, we evaluated the model's sensitivity to the number of experts. Specifically, we varied total number of experts and selected half of them, with corresponding results presented in Figure 6 (a). In ( we fixed the total number of experts at 10 while varying the number of selected experts. The findings indicate that choosing an optimal subset of experts, rather than an excessive or minimal number, enhances the model's performance.

---

Micro 11001110 pM
Macro F1
Mcro F1 (%))
5
(a)
(b)
Figure 5: The classification performance of MvCGE w.r.t. hyperparameters a and on the (a) ACM dataset and (b) YELP dataset.
83
Micro (1g
Micro (%g
F1 29
80
1
22
37
#Selectad/ Total Experts
#Selected/ Total Experts
# #Selected Experts
#Selected Experts
(a)
(b)
Figure 6: The classification performance of MvCGE w.r.t. (a) the number of selected and total graph experts and (b) the number of selected graph experts on the YELP dataset.
Conclusion
In this paper, address the critical challenge of balancing consistency and complementarity in multi-view graph learning. Traditional multi-view GNNs often struggle to adapt to the heterogeneous structural properties of multi-view graphs due rigid weight-sharing designs. To overcome these limitations, we propose MvCGE, novel framework inspired by Mix- ture of Experts, which introduces layer-wise collaborative graph experts to dynamically process multi-view graphs while preserving both shared and view-specific information. The integration of load equilibrium loss and graph discrepancy loss effectively mitigates model collapse and enhances the discriminative power of learned representations. Extensive experiments across diverse datasets demonstrate that MvCGE achieves state-of-the-ard performance, showcasing its flexibility and robust- ness in handling complex multi-view graph data. One possible limitation is that we do not discuss the homophily and heterophily of graphs, although we adopt both types of graphs in our experiments.
isolated multi-channel
6
Acknowledgements
This work was supported by the National Natural Science Foundation of and 62372408).
China (Grant Nos. 62202422
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Reviews:
[{"review_id": "oychJdTQFk", "reviewer": "Reviewer_DaD6", "summary": "This paper identifies that existing multi-view GNNs either share parameters across views (over-emphasizing consistency but losing view-specific nuance) or allocate separate channels per view (preserving complementarity but sacrificing inter-view synergy). They propose MvCGE, which at each layer maintains a shared pool of graph-convolution experts and uses a graph-aware top-K routing to dynamically assign each view’s nodes to a subset of experts. A load-equilibrium loss ensures no expert is neglected, and a graph-discrepancy loss (via MMD) enforces that different experts learn distinct representations. Experiments on five multi-view and five single-view citation/dataset benchmarks show MvCGE consistently outperforms strong baselines in Macro-F1 and Micro-F1.", "questions": "1.  Scalability: How does inference time grow with graph size and expert count?\n2.  How to define the view of multi-view graphs, is there any different designs of different experts ? Would different experts receive graph inputs only from that views, are different combinationn of designs have impacts towards final results ?\n3.  Expert specialization: Have you observed that particular experts consistently specialize on certain views or substructures? Can these specializations be interpreted?\n4.   Robustness to view noise: If one view is very noisy or missing, how does the routing mechanism compensate?\n5.    Loss weighting: How sensitive is performance to the choice of α (load-equilibrium) and β (discrepancy)? Is there a principled way to set them?\n6.    Generalization to other tasks: Can the same MvCGE framework be applied to link prediction or graph regression problems? What changes would be needed?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n- MOE for  multi-view GNN learning\nBy decoupling expert count from view count and dynamically routing, MvCGE achieves both architectural consistency (shared experts) and view complementarity (expert specialization) in a unified framework.\n-   Novel Regularizers\nThe load-equilibrium loss prevents expert collapse (ensuring each expert is used), and the graph-discrepancy loss leverages MMD to keep expert representations distinct. \n-   Strong Empirical Gains \nOn five multi-view graphs (e.g., ACM-M, YELP), MvCGE achieves good performance compared to other baselines, and it also improves over state-of-the-art single-view GNNs on standard benchmarks.\n\n**Weaknesses**\n- The Ldsp loss:\n This loss is to minimize the discrepancy between distributions from different graph experts. THIS DESIGN is actually against the orginal design to perform distinct filtering. Thus, this loss is questionable.  \n- Complexity issues:\n   This solution introduce considerable complexity which might not be effective on large scale graphs. The authors fail to provide the complexity analysis and the trainning/execution time of the proposed solution.  The Top-K gating per node incurs extra computation and may be hard to scale to very large graphs or high expert counts without further efficiency tricks.\n● Limited Task Scope\n    Evaluation focuses solely on semi-supervised node classification. It remains unclear how MvCGE would perform on regression, link prediction, or graph-level tasks.\n● Lack of reproducity, Source code and Settings are not provided\n     This paper provide no source code with very complex design. The detailed settings are also missing, e.g. how to generate multi-view for single-view graphs , how to define the feature similarity threshold for connections? Settings on designs of differnt experts, number of experts, topK are not mentioned.", "was_revised": true, "final_justification": "Thanks for the detailed reply. It resolve most of my concerns. I would like to raise my score accordingly.", "rebuttal": "We appreciate your constructive feedback and your recognition of our work. Many of your comments provide valuable insights that deserve further exploration. In the following, we systematically address each weakness (W) and question (Q) raised, and we would like to clarify several possible and potential misunderstandings.\n\n**W1. Confusion about $\\mathcal{L}\\_{\\mathrm{dsp}}$**\n\n**Response:** We regret any confusion that may have occurred and would like to provide some clarifications. As mentioned at lines 239-240, the Maximum Mean Discrepancy (MMD) $\\mathcal{M}^2_{\\mathcal{H}}(\\mathbb{P}\\_{\\mathcal{Q}}, \\mathbb{P}\\_{\\mathcal{Q}'}) $ [1] is adopted to quantify the **discrepancy** between different graph experts. Then Eqn. (18) gives the MMD-based loss function:  \n$$\n\\mathcal{L}\\_{\\mathrm{dsp}} = -\\frac{1}{L}\\sum\\_l \\frac{1}{M(M-1)} \\sum\\_{m<m'}\\mathcal{M}\\^2\\_{\\mathcal{H}}\\Big(\\mathbb{P}\\_{\\mathcal{Q}^{(l)}\\_m}, \\mathbb{P}\\_{\\mathcal{Q}^{(l)}\\_{m'}}\\Big).\n$$\nTherefore, minimizing $ \\mathcal{L}\\_{\\mathrm{dsp}} $ ***maximizes*** the differences among graph filters, which is ***consistent with our original design*** that promotes MvCGE to capture complementary information. We will add explicit statements like \"minimizing the total loss maximizes MMD, thereby increasing inter-expert diversity\" to avoid potential misunderstandings.  \n\n\n**W2/Q1. Complexity concerns**\n\n**Response:** Our solution introduces a minor additional overhead but ***does not lead to a considerable increase in overall complexity*** [2, 3]. MoE has been widely adopted in LLMs precisely because it allows for the expansion of the model capacity while maintaining inference efficiency through sparse expert activates [4, 5]. In this work, we provide a detailed theoretical complexity analysis in ***Appendix B***. When employing $ E $ experts, the computational complexity of the gating mechanism within a single MvCGE layer is $ \\mathcal{O}(NDE+|\\mathcal{E}|D) $, while that of each hierarchical graph expert (e.g., GCN) is $ \\mathcal{O}(ND^2+|\\mathcal{E}|D) $. In practice, $ E $ is typically not much larger than the $ D $, so it will not be a bottleneck. The most time-consuming operation remains the graph convolution ($ \\mathcal{O}(|\\mathcal{E}|D) $). It is the gating mechanism that allows each sample to activate only a small subset of $K$ experts, even with dozens of graph experts. In addition, for spectral GNNs such as GCN, we can decouple the filter computation from the learnable parameters [6, 7], substantially reducing the number of filtering operations required. Empirically, Appendix A reports the results of MvCGE on two large-scale graphs, and here we further provide efficiency comparisons on three datasets to offer a clearer perspective:\n\n| Method | ACM (Time) | ACM (Memory) | Citeseer (Time) | Citeseer (Memory) | Flickr (Time) | Flickr (Memory) | UAI (Time) | UAI (Memory) |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| **GCN** | 1.82s | 24.49 MB | 1.82s | 51.04 MB | 3.01s | 369.76 MB | 1.96s | 65.28 MB |\n| **MvCGE (3 experts, top 1)** | 3.60s | 25.41 MB | 3.33s | 52.21 MB | 4.91s | 374.16 MB | 3.52s | 67.00 MB |\n| **MvCGE (6 experts, top 2)** | 4.95s | 25.57 MB | 5.07s | 52.39 MB | 7.03s | 374.54 MB | 5.46s | 67.20 MB |\n\nWe evaluated MvCGE under two settings: (1) 3 experts with each sample selecting the top-1 expert, and (2) 6 experts with each sample selecting the top-2 experts. The experiments were conducted on single-view graphs. Note that GCN processes only a single view, whereas MvCGE processes two views simultaneously. Nevertheless, MvCGE did not introduce significant computational overhead, which is consistent with our theoretical analysis and other empirical results.\n\n**W3/Q6. Generalization to other tasks**\n\n**Response:** In this work, we primarily validate the effectiveness of our model on the common graph learning task, semi-supervised node classification, rather than positioning MvCGE as a foundation model. As we demonstrated, it is straightforward to adapt the model to some graph tasks by simply replacing $ \\mathcal{L}\\_{\\text{task}} $. We have conducted additional experiments on link prediction and graph-level classification, as shown below:\n\n| Method | IMDB-B | MUTAG | PTC |\n| --- | --- | --- | --- |\n| PATCHYSAN | 71.0 ± 2.2 | 92.6 ± 4.2 | 60.0 ± 4.8 |\n| DGCNN | 70 | 85.8 | 58.6 |\n| AWL | 74.5 ± 5.9 | 87.9 ± 9.8 | - |\n| MLP | 73.7 ± 3.7 | 84.0 ± 6.1 | 66.6 ± 6.9 |\n| GIN | 75.1 ± 5.1 | 89.4 ± 5.6 | 64.6 ± 7.0 |\n| GCN | 74.0 ± 3.4 | 85.6 ± 5.8 | 64.2 ± 4.3 |\n| GraphSAGE | 72.3 ± 5.3 | - | 63.9 ± 7.7 |\n| Ours | 76.5 ± 2.4 | 91.2 ± 3.9 | 68.6 ± 3.4 |\n\n\n| Category | Models | Cora MRR | Cora AUC |\n| --- | --- | --- | --- |\n| GNN | GCN | 32.5 ± 6.9 | 95.0 ± 0.3 |\n|   | GAT | 31.9 ± 6.1 | 93.9 ± 0.3 |\n|   | SAGE | 37.8 ± 7.8 | 95.6 ± 0.3 |\n|   | GAE | 30.0 ± 3.2 | 95.1 ± 0.3 |\n| Pairwise Info | SEAL | 26.7 ± 5.9 | 90.6 ± 0.8 |\n|    | BUDDY | 26.4 ± 4.4 | 95.1 ± 0.4 |\n|    | Neo-GNN | 22.7 ± 2.6 | 93.7 ± 0.4 |\n|    | NCN | 32.9 ± 6.3 | 96.8 ± 0.2 |\n|    | NCNC | 29.8 ± 5.4 | 96.7 ± 0.3 |\n|    | NBFNet | 37.7 ± 4.0 | 96.9 ± 0.3 |\n|    | PEG | 22.8 ± 1.8 | 94.5 ± 0.4 |\n| Ours | MvCGE | 36.3 ± 1.6 | 96.6 ± 0.3 |\n\nFor the graph-level tasks, we followed the experimental settings of [8], while for the link prediction experiments, we adopted the settings from [9]. As can be seen, MvCGE continues to demonstrate strong performance on these tasks. We will consider evaluating the model in even more scenarios in future work. Thank you for your valuable suggestion.\n\n\n\n**W4/Q5. Reproducibility**\n\n**Response:** We clarify that we have included the anonymous repository link and hardware details in ***Appendix A.2***, which ensures reproducibility of our work. Here we further provide the detailed settings to address your concern: \n- For single-view graphs, we generate an additional feature-similarity $ k $NN graph ($ k=10 $, Euclidean distance) as the second view.  \n- For the number of experts/Top-K, we fix them as $E=6$ and $K=2$ for Table 1 and 2, and analyze the impact of their choice in our paper (refer to Fig. 6), and we will add clear statements of the exact $ k $NN threshold and other tuning grids to an Implementation Details subsection.\n- Regarding the sensitivity for $ \\alpha $ and $ \\beta $, we have analyzed in the main body of our submission (refer to Fig. 5). As can be observed, performance is stable in $ \\pm1 $ order-of-magnitude around the chosen $ \\alpha $ and $ \\beta $. We will revise our paper to include more detailed settings for reproducibility. \n\n**Q2. View definition & expert design**\n\n**Response:** For real multi-view datasets we use the provided edge sets. For single-view datasets we add a feature-similarity kNN graph (see above). All experts share the same base spectral GNN layer (e.g., GCN or ChebNet ). Diversity is imposed mainly by the mechanism of MoE and our proposed loss $\\mathcal{L}\\_{\\text{dsp}}$.\n\n\n**Q3. Expert specialisation**\n\n**Response:** After training, it is indeed observed that different experts tend to focus on samples from a specific view. This forms the unique inductive bias of MvCGE, enabling the preservation of view-specific information. Regarding the substructures you have mentioned, since MvCGE is built upon graph spectral theory, different experts are able to learn distinct filter parameters, thereby capturing homophilic/heterophilic patterns. We have conducted visualization experiments on expert specialization and routing. Due to the NeurIPS 2025 rebuttal policy, we will include detailed discussion and related results in future versions.  \n\n**Q4. Robustness to noise**\n \n**Response:** Owing to the presence of the multi-view aggregation module, MvCGE is theoretically capable of assigning low weights or even blocking those views that result in poor training loss (when using gating-based aggregation). As a result, the framework exhibits a certain degree of tolerance to noise. We conduct an experiment on ACM for a better understanding:\n| # Noisy Edges    | MaF1              | MiF1              |\n|-------------|-----------------|------------------|\n| 0.01\\|𝓔\\|   | 86.6 ± 0.4      | 86.6 ± 0.4      |\n| 0.05\\|𝓔\\|   | 85.2 ± 0.8      | 85.2 ± 0.8      |\n| 0.1\\|𝓔\\|    | 83.5 ± 0.5      | 83.5 ± 0.5      |\n| 0.5\\|𝓔\\|    | 81.7 ± 3.2      | 81.8 ± 3.2      |\n| \\|𝓔\\|       | 79.4 ± 5.4      | 79.7 ± 5.4      |\n\nHere, we construct a noisy view by randomly generating an adjacency matrix and test MvCGE's performance under different noise rates. We will include additional experiments on robustness to further discuss this aspect.\n\n```\n[1] A kernel two-sample test. JMLR 2012.\n[2] Towards understanding the mixture-of-experts layer in deep learning. NeurIPS 2023.\n[3] Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. ICLR 2017.\n[4] Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity. JMLR 2022.\n[5] Scaling vision with sparse mixture of experts. NeurIPS 2021.\n[6] Semi-Supervised Classification with Graph Convolutional Networks. ICLR 2017.\n[7] Convolutional neural networks on graphs with chebyshev approximation, revisited. NeurIPS 2022.\n[8] How powerful are graph neural networks? ICLR 2018.\n[9] Evaluating graph neural networks for link prediction: Current pitfalls and new benchmarking. NeurIPS 2023.\n```"}, {"review_id": "pzSscsmIlF", "reviewer": "Reviewer_HqvP", "summary": "This paper addresses the challenge of balancing consistency and complementarity in multi-view graph neural networks (MvGNNs) by introducing a Mixture-of-Experts–inspired framework called Multi-view Collaborative Graph Mixture of Experts (MvCGE). At each layer, a set of shared “graph experts” (i.e., distinct GNN filters) is dynamically routed per view via a graph-aware top-k gating mechanism. To prevent expert collapse and encourage diversity, the authors also propose two losses. Experiments on five multi-view and six single-view benchmarks demonstrate that MvCGE consistently outperforms baselines.", "questions": "Q1. How exactly is the routing function calculated? Equations 8, 10 and 11 are a bit confusing, can you provide a step-by-step description or a table of algorithm for better clarity?\n\nQ2. In Equation 13, why does the superscript $(l)$ disappear and why is it inconsistent with Equation 8?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nS1. Novel and well-motivated. The framework innovatively addresses the key challenge of balancing consistency and complementarity in multi-view learning by combining MoE with GNN, has a practical impact and makes sense for the community.\n\nS2. It is interesting that the work designs the graph discrepancy and the graph-aware router, which are capable of regularizing the experts and learning awareness of graph structures.\n\nS3. The evaluation of the proposed method is comprehensive, experiments include performance comparisons over both single-view and multi-view graphs, sensitivity analyses, and visualization.\n\n**Weaknesses**\n\nW1. Reproducibility is doubtful. The authors have included only an appendix in the supplemental material and have not provided any source code or any form of pseudo-code for reproduction. This may affect the credibility of the experimental results.\n\nW2. The Methodology section is hard to follow. The complex notations made me spend more time reading about the methodology, especially for the collaborative experts and graph discrepancy regularization. I list the questions in the following part.\n\nW3. Lack of some literature. In related work, the authors only introduce a few MoE-GNN papers, and some recent works are not discussed or compared as baselines, leaving unclear how this work positions itself in the literature.\n\nW4. Limited theoretical analysis. Beyond intuitive motivation, the paper lacks a formal or theoretical analysis of the proposed method.", "was_revised": true, "final_justification": "The authors have addressed my concerns regarding reproducibility and readability. The paper is novel and technically sound. I am raising my score and recommend acceptance.", "rebuttal": "We appreciate your thorough review and insightful comments. Below we address every weakness (W) and question (Q) in detail, and we explain the concrete additions that will appear in our revised manuscript.\n\n**W1. Reproducibility**\n\n**Response:** To facilitate online access without the need to download additional files, we did not include the source code directly in the supplementary materials. Instead, we have provided an anonymous link for convenient viewing. We would like to clarify that the anonymous project link to the source code, as well as the implementation platform details, are both ***already included in Appendix A.2***. We believe these materials sufficiently demonstrate the effectiveness of the proposed MvCGE and ensure the reproducibility of the experiments presented in this paper.\n\n**W2/Q1. Details about the method**\n\n**Response:** We apologize for any inconvenience this may have caused. To ensure precise expression, we adopted a relatively complex notation system. In future versions, we will adjust the organization of the main text and appendices by simplifying the notation as much as possible in the main body, moving certain technical details to the appendix, and providing comprehensive explanations for each symbol. This will help balance readability and accuracy. Here we provide a step-by-step description of the routing function:\n\n| Step                          | Formula / code                                                                                               | Explanation                                               |\n| ----------------------------- | ------------------------------------------------------------------------------------------------------------ | --------------------------------------------------------- |\n| 1. **Aggregation** | $u_i = [P_v H^{(l)}_v]_i$                                                                                    | Graph convolution of node *i* under view *v*.             |\n| 2. **Raw scores**             | $s_i = u_i W_{\\text{gate}}$                                                                                  | Linear map to E-dimensional gate scores.           |\n| 3. **Top-K sparsification**          | $s_i \\leftarrow s_i \\cdot \\mathbf 1_{\\{s_i \\ge \\delta\\}}$  ($\\delta = K$-th largest value)                                | Keeps only the $K$ strongest expert. |\n| 4. **Softmax**                | $[\\Gamma^{(l)}\\_v]\\_{i,:} = \\text{softmax}(s\\_i)$                                           | Normalises surviving scores.                              |\n| 5. **Expert fusion**          | $h\\^{(l+1)}\\_{v,i} = \\sum\\_m [\\Gamma^{(l)}\\_v]_{i,m} \\, f^{(l)}\\_{\\Theta\\_m}(G\\_v,H^{(l)}\\_v)_i$   | Weighted sum yields node output.                          |\n\n**W3. Lack of GNN-MoE-based literature**\n\n**Response:** As discussed in the related work section, the MoE [1] architecture has been widely adopted in many modern large language models (LLMs) [2], yet its potential in graph learning remains largely underexplored. A few recent studies have proposed MoE-based methods for graph learning, but these primarily focus on single-view graphs [3, 4, 5]. In contrast, the MoE architecture is inherently well-suited for modeling complex multi-view graphs, which represents a clear research gap. Consequently, we were unable to identify directly comparable baselines in the literature. In this work, we adopt the classical single-view GMoE [3] as a baseline for comparison. We conduct the comparison on both single- and multi-view datasets:\n\n| Method   | ACM MaF1     | ACM MiF1     | DBLP MaF1    | DBLP MiF1    |\n|----------|--------------|--------------|--------------|--------------|\n| GCN      | 0.7860       | 0.7881       | 0.9012       | 0.9163       |\n| DGI      | 0.2235       | 0.3690       | 0.2432       | 0.3763       |\n| HAN      | 0.9147       | 0.9139       | 0.8927       | 0.9039       |\n| DMGI     | 0.8666       | 0.8681       | 0.6567       | 0.7106       |\n| IGNN     | 0.8290       | 0.8270       | 0.8681       | 0.8750       |\n| MRGCN   | 0.8758       | 0.8745       | 0.9007       | 0.9047       |\n| SSDCM    | 0.8765       | 0.8763       | 0.8942       | 0.8990       |\n| MHGCN    | 0.8887       | 0.8907       | 0.9300  | 0.9360   |\n| AMOGCN   | 0.9241  | 0.9238   | 0.9227       | 0.9280       |\n| HMGE     | 0.9080       | 0.9066       | 0.9156       | 0.9235       |\n| GMoE     |  0.8816  | 0.8808  | 0.8736  | 0.8858  |\n| MvCGE    | 0.9348   | 0.9245   | 0.9256   | 0.9317   |\n\n| Method   | ACM MaF1   | ACM MiF1   | Citeseer MaF1 | Citeseer MiF1 |\n|----------|------------|------------|---------------|---------------|\n| GCN      | 0.8827     | 0.8842     | 0.6523        | 0.6909        |\n| SGC      | 0.8116     | 0.8087     | 0.6157        | 0.6642        |\n| APPNP    | 0.8837     | 0.8824     | 0.6510        | 0.6972        |\n| JKNet    | 0.8524     | 0.8545     | 0.6813        | 0.7292        |\n| DAGNN    | 0.8699     | 0.8917     | 0.6812        | 0.7290        |\n| GCNIi    | 0.8978     | 0.8980     | 0.6712        | 0.7064        |\n| GNNHF    | 0.9052     | 0.9067     | 0.6760        | 0.7249        |\n| AMGNN    | 0.8996     | 0.8993     | 0.6873        | 0.7292        |\n| HiDNet   | 0.8958     | 0.8961     | 0.6661        | 0.7146        |\n| AGNN     | 0.8973     | 0.8974     | 0.6264        | 0.6721        |\n| GMoE     | 0.8964  | 0.8950 | 0.6789  | 0.7107  |\n| MvCGE    | 0.9110 | 0.9114 | 0.6908    | 0.7356    |\n\nAs shown by the experimental results, GMoE demonstrates competitive performance on both types of datasets. However, it consistently falls short of MvCGE, further indicating that while the MoE architecture holds significant potential in multi-view graph scenarios, existing frameworks are insufficient to fully exploit this advantage.\n\n**W4. Limited theoretical analysis**\n\n**Response:** Although we do not provide a direct theoretical analysis of the proposed method, we do offer theoretical justification for its rationality. Specifically, a formal lower-bound analysis of single-filter transferability across views is presented in Appendix C. This analysis theoretically demonstrates that a single graph filter cannot adapt to multiple graph structures with distinct frequency characteristics, whereas a combination of multiple filters does not suffer from this limitation. Building upon this insight, we considered the balance between consistency and complementarity, which motivated the design of the MvCGE framework.\n\n**Q2. Regarding Equation (13)**\n\n**Response:** The section containing Equation (13) introduces the graph discrepancy loss, which measures the distributional differences between any pair of graph experts within each layer of MvCGE. Therefore, for the sake of clarity and conciseness, we omit the superscript $(l)$ and use $f_{\\Theta_m}$ and $f_{\\Theta_{m'}}$ to denote any two experts within the same layer when describing this loss.\n\n```\n[1] Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. ICLR 2017.\n[2] Deepseek-v3 technical report. arXiv preprint arXiv:2412.19437 (2024).\n[3] Graph mixture of experts: Learning on large-scale graphs with explicit diversity modeling. NeurIPS 2023.\n[4] Mixture of experts for node classification. ICMR 2025.\n[5] Fair graph representation learning via diverse mixture-of-experts. WWW 2023.\n```"}, {"review_id": "GZNuUg5Q44", "reviewer": "Reviewer_zXFJ", "summary": "This paper proposes a MoE-inspired framework to harmonize consistency and complementarity in multi-view graph learning. By employing shared collaborative graph experts and a graph-aware router, the model dynamically selects experts per view while maintaining architectural consistency. Two regularization strategies, load balancing and distributional divergence, prevent expert dominance and enhance view-specific representations. Experiments show state-of-the-art performance across 10 datasets.", "questions": "- How does homophily/heterophily interact with the proposed model?\n- What do proposition 1 and corollary 1 suggest?\n- See weaknesses for some questions.", "limitations": "Yes the authors discuss limitations in the conclusion.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strength\n\n- The paper is well-motivated and well-written. The authors do identify a critical challenge in multi-view graph learning and the design limitations of existing multi-view graph neural networks. The design principles of this work could inspire MoE applications in other graph tasks.\n- One principal strength of this work is the proposed architecture decouples expert count from view count, which achieves dynamic adaptation to multi-view graphs and without composing much computational burden.\n- Providing executable source code is encouraged, I run the code and can reproduce the reported results. BTW, the code is well-organized and easy to read.\n\n## Weakness\n\n- As the author mentioned in the conclusion, one limitation of this paper is the absence of a discussion on the interaction between homophily/heterophily and the proposed MvCGE. Addressing this issue may provide more insights into this work.\n- I note that the authors provide some theoretical analysis in Appendix C, can the authors clarify the necessity of Proposition 1 and Corollary 1 and how they demonstrate the validity of the proposed approach?\n- The discussion of graph discrepancy appears to lack sufficient definitional clarity. Regarding the distributions $\\mathbb{P}\\_{Q}$ and $\\mathbb{P}\\_{Q’}$, what exact probability measures do $\\mathbb{P}\\_{Q}$ and $\\mathbb{P}\\_{Q’}$ represent in the proposed framework? How to sample instances from these distributions to compute the MMD-based graph discrepancy $M_{H}(\\mathbb{P}\\_{Q}, \\mathbb{P}\\_{Q’})$?", "was_revised": true, "final_justification": "This work presents both novelty and clear presentation, and it addresses a practical and important problem. This work is expected to have a positive impact on the graph learning community. The main concerns were related to the explanation of certain issues, most of which have been addressed in the authors' response. Therefore, I maintain the high score.", "rebuttal": "We sincerely thank the reviewer for their careful reading and recognition of our work. Below, we provide point-by-point responses to address your concerns.\n\n**W1/Q1. Discussion on homophily/heterophily**\n\n**Response:** In this work, we did not explicitly discuss the issue of homophily or heterophily. However, spectral GNNs are closely related to these properties [1, 2]. In homophilic graphs, neighboring nodes tend to share similar attributes or labels, implying that the energy is concentrated in the low-frequency components of the Laplacian spectrum [3, 4]. Conversely, heterophilic graphs exhibit substantial variations between nodes, leading to more high-frequency components. In multi-view graphs, despite sharing the same node set, different views may exhibit significant differences in homophily and heterophily, resulting in distinct spectral distributions. This heterogeneity renders single graph convolutions/filters inadequate for unified processing, which aligns with our theoretical motivation. For instance, the YELP dataset demonstrates substantial inter-view divergence, which explains why MvCGE achieves notably superior performance on it. We will strengthen the discussion of these aspects in the Method and Experiment sections.\n\n**W2/Q2. Further discussions on theoretical analysis**\n\n**Response:** Proposition 1 and Corollary 1 primarily serve as the theoretical motivation to expose the inherent limitations of shared graph filters in multi-view graph learning. These insights motivate us to explore collaboration graph filter and propose our solution: By designing a team of specialized graph filters under the MoE framework, MvCGE enables: (1) View-Specific Filter Selection: Each view dynamically activates the most suitable filter combination through graph-aware routing; (2)Collaborative Spectral Processing: Distinct filters specialize in handling complementary frequency bands across views. Therefore, the total error can be decomposed and now depends on the optimal combination that can be chosen for each view from all graph experts, circumventing the error growth bound in Proposition 1. We will reconsider the placement and presentation of these theoretical analyses in future versions.\n\n**W3. Graph discrepancy loss clarification**\n\n**Response:** In MvCGE, each graph expert is assigned a set of samples and generates node representations for them, and we expect these graph experts to learn different latent spaces during the training process, and thus need to measure the distribution differences between the learned representations among the graph experts. We build on research related to graph isomorphism computation [5] by introducing the readout function combined with graph experts to obtain graph embeddings, which are essentially global representations that encode graph structure. And the distribution of graph embeddings is used to measure the difference between graph experts. Therefore, we first perform a readout of the representations learned by each graph expert for a certain view by $\\mathbf{z} = \\mathcal{P}(\\mathbf{Z})$.\nSubsequently, using $Q$ to represent the combination of the readout function and the graph expert, we treat the set of all these embeddings (across different views) as samples from the distribution $\\mathbb{P}\\_{Q}$. Hence, $\\mathbb{P}\\_{Q}$ is the empirical distribution over the graph-level embeddings ${\\mathbf{z}}$. Whereas different graph experts have different distributions, denoting as $\\mathbb{P}\\_{Q}$ and $\\mathbb{P}\\_{Q'}$.\nFinally, we measure how different the distributions are by using the powerful Maximum Mean Discrepancy (MMD) with an RBF kernel. For a certain view, we compute the node embeddings via graph expert $m$, apply the readout function $\\mathcal{P}$, and obtain a single embedding $\\mathbf{z}$. Doing this for each view in the dataset yields a set of $\\mathbf{z}$, which we treat as samples from $\\mathbb{P}\\_{Q}$, and $\\mathbb{P}\\_{Q}$ is simply the empirical distribution of these embeddings.\n\n```\n[1] Graph neural networks for graphs with heterophily: A survey. arXiv preprint arXiv:2202.07082 (2022).\n[2] Addressing heterophily in graph anomaly detection: A perspective of graph spectrum. WWW 2023.\n[3] Revisiting heterophily for graph neural networks. NeurIPS 2022.\n[4] Beyond low-frequency information in graph convolutional networks. AAAI 2021.\n[5] How powerful are graph neural networks? ICLR 2018.\n```"}, {"review_id": "pLHdKFbOft", "reviewer": "Reviewer_7FN5", "summary": "This paper introduces Multi-view Collaborative Graph Mixture of Experts (MvCGE), a novel framework that unifies multi-view learning and graph neural networks. MvCGE employs a pool of layer-wise graph experts that are dynamically routed to each view via a graph-aware top-K gating mechanism. Consistency is enforced by sharing expert parameters across views, while complementarity is preserved through two regularizers. Extensive experiments on five multi-view and six single-view benchmarks demonstrate consistent performance gains over strong baselines.", "questions": "Does the combination of multiple GNNs and multiple graphs result in a high computational burden? If so, does the proposed approach include any improvement for scalability?", "limitations": "Yes. The authors note the omission of homophily/heterophily discussion. I recommend the authors to include a discussion (or empirical analysis) of how their method performs under varying graph homophily levels.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. The work is clearly motivated by the trade-off between consistency and complementarity in multi-view GNNs.  \n2. The proposed framework seems to be novel and bridge the research gap in multi-graph learning.  \n3. Most of the descriptions and formalizations in the methods are clear, and it is convinced that the proposed MoE-based framework can be a good solution.  \n4. Evaluation across ten datasets (including large-scale Freebase and arXiv in the appendix) with both multi-view and single-view settings demonstrates broad applicability.\n\nWeaknesses:\n\n1. As per my understanding, introducing multiple GNNs on multiple graphs will increase the runtime and memory footprint by a significant amount compared to the baselines. However, the authors claim that the proposed method is efficient and does not impose a significant computational burden. The authors should clarify this issue.  \n2. The model requires tuning on several hyperparameters $\\alpha$, $\\beta$, total experts $M$, and top-$k$. although explored their impact, the need to tune four interdependent hyperparameters may hinder practical adoption.  \n3. Some equations (e.g., the generalized MMD in Eq. 16 vs. 17) and definitions (routing mechanism) could be streamlined in the main paper and elaborated in the appendix, or readers may struggle with these.", "was_revised": true, "final_justification": "Most of my concerns have been addressed, and therefore I decide to maintain my score.", "rebuttal": "**W1/Q1. Complexity concerns**\n\n**Response:** Our solution introduces a minor additional overhead but does not lead to a considerable increase in overall complexity [1, 2]. MoE has been widely adopted in LLMs precisely because it allows for the expansion of the model capacity while maintaining inference efficiency through sparse expert activates [3, 4]. In this work, we provide a detailed theoretical complexity analysis in Appendix B. When employing $ E $ experts, the computational complexity of the gating mechanism within a single MvCGE layer is $ \\mathcal{O}(NDE+|\\mathcal{E}|D) $, while that of each hierarchical graph expert (e.g., GCN) is $ \\mathcal{O}(ND^2+|\\mathcal{E}|D) $. In practice, $ E $ is typically not much larger than the $ D $, so it will not be a bottleneck. The most time-consuming operation remains the graph convolution ($ \\mathcal{O}(|\\mathcal{E}|D) $). It is the gating mechanism that allows each sample to activate only a small subset of $K$ experts, even with dozens of graph experts. In addition, for spectral GNNs such as GCN, we can decouple the filter computation from the learnable parameters [5, 6], substantially reducing the number of filtering operations required. Empirically, Appendix A reports the results of MvCGE on two large-scale graphs, and here we further provide efficiency comparisons on three datasets to offer a clearer perspective:\n\n| Method | ACM (Time) | ACM (Memory) | Citeseer (Time) | Citeseer (Memory) | Flickr (Time) | Flickr (Memory) | UAI (Time) | UAI (Memory) |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| **GCN** | 1.82s | 24.49 MB | 1.82s | 51.04 MB | 3.01s | 369.76 MB | 1.96s | 65.28 MB |\n| **MvCGE (3 experts, top 1)** | 3.60s | 25.41 MB | 3.33s | 52.21 MB | 4.91s | 374.16 MB | 3.52s | 67.00 MB |\n| **MvCGE (6 experts, top 2)** | 4.95s | 25.57 MB | 5.07s | 52.39 MB | 7.03s | 374.54 MB | 5.46s | 67.20 MB |\n\n**Response:** We evaluated MvCGE under two settings: (1) 3 experts with each sample selecting the top-1 expert, and (2) 6 experts with each sample selecting the top-2 experts. The experiments were conducted on single-view graphs. Note that GCN processes only a single view, whereas MvCGE processes two views simultaneously. Nevertheless, MvCGE did not introduce significant computational overhead, which is consistent with our theoretical analysis and other empirical results.\n\n**W2. Multiple hyperparameters tuning**\n\n**Response:** Although the proposed model does require tuning of four hyperparameters, in practice, we fixed the number of experts to 6 and the number of selected experts to 2, and still achieved strong performance across all datasets. Furthermore, as demonstrated in the experiments shown in Fig. 6, it is not necessary to finely tune these two hyperparameters within a reasonable range.\n\n**W3. Regarding equations and definitions**\n\n**Response:** We apologize for any inconvenience caused by the complex notation and formulas. We will carefully consider your suggestions and systematically adjust the organization and presentation of the text to facilitate readers’ understanding of our method.\n\n```\n[1] Towards understanding the mixture-of-experts layer in deep learning. NeurIPS 2023.\n[2] Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. ICLR 2017.\n[3] Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity. JMLR 2022.\n[4] Scaling vision with sparse mixture of experts. NeurIPS 2021.\n[5] Semi-Supervised Classification with Graph Convolutional Networks. ICLR 2017.\n[6] Convolutional neural networks on graphs with chebyshev approximation, revisited. NeurIPS 2022.\n```"}]

Author rebuttal:

Meta-review: This paper proposes a novel and well-motivated mixture-of-experts framework for multi-view GNNs, with clear motivation and strong empirical validation. Concerns regarding scalability, notation clarity, and theoretical depth were raised by reviewers, but the authors’ rebuttal addressed these points. All reviewers expressed satisfaction with the clarifications and advocated for acceptance. In summary, I agree with the reviewers and recommend accepting this paper.
Decision: Accept (poster)