Less but More: Linear Adaptive Graph Learning Empowering Spatiotemporal Forecasting
Authors: Jiaming Ma
Binwu Wang
Guanjun Wang
Kuo Yang
Zhengyang Zhou
Pengkun Wang
Xu Wang
Yang Wang
Venue: NeurIPS 2025 poster ()
Keywords: Spatiotemporal learning; Graph Representation Learning; Time Series; Deep Learning
Primary area: deep_learning

Abstract: The effectiveness of Spatiotemporal Graph Neural Networks (STGNNs) critically hinges on the quality of the underlying graph topology. While end-to-end adaptive graph learning methods have demonstrated promising results in capturing latent spatiotemporal dependencies, they often suffer from high computational complexity and limited expressive capacity. In this paper, we propose MAGE for efficient spatiotemporal forecasting. We first conduct a theoretical analysis demonstrating that the ReLU activation function employed in existing methods amplifies edge-level noise during graph topology learning, thereby compromising the fidelity of the learned graph structures. To enhance model expressiveness, we introduce a sparse yet balanced mixture-of-experts strategy, where each expert perceives the unique underlying graph through kernel-based functions and operates with linear complexity relative to the number of nodes. The sparsity mechanism ensures that each node interacts exclusively with compatible experts, while the balancing mechanism promotes uniform activation across all experts, enabling diverse and adaptive graph representations. Furthermore, we theoretically establish that a single graph convolution using the learned graph in MAGE is mathematically equivalent to multiple convolutional steps under conventional graphs. We evaluate MAGE against advanced baselines on multiple real-world spatiotemporal datasets. MAGE achieves competitive performance while maintaining strong computational efficiency.

Less but More: Linear Adaptive Graph Learning Empowering Spatiotemporal Forecasting
Jiaming Ma', Binwu Wang Guanjun Wang', Kuo Yang' Zhengyang ?2 Pengkun Wang Xu Wang Yang 12n University of Science and Technology of China (USTC), Hefei, Anhui, China Suzhou Institute for Advanced Research, USTC, Suzhou, Jiangsu, China {Ji diimmigMa always, yangkuu)@mail ust.. edu. (wbw2024, zzy0929, pengkun, wx309, angyan)@ustc.com wustu.uutt.uuuu edu.cn
Abstract
The effectiveness of Spatiotemporal Graph Neural Networks (STGNNs) critically hinges on the quality of the underlying graph topology. While end-to-end adaptive graph learning methods have demonstrated promising results in capturing latent spatiotemporal dependencies, they often suffer from high computational complexity and limited expressive capacity. In this paper, we propose MAGE for efficient spatiotemporal forecasting. We first conduct a theoretical analysis demonstrating that the ReLU activation function employed in existing methods amplifies edge- level noise during graph topology learning, thereby compromising the fidelity of the learned graph structures. To enhance model expressiveness, we introduce a sparse yet balanced mfxture-of-experts strategy, where each expert perceives the unique underlying graph through kernel-based functions and operates with linear complexity relative to the number of nodes. The sparsity mechanism ensures that each node interacts exclusively with compatible experts, while the balancing mechanism promotes uniform activation across all experts, enabling diverse and adaptive graph representations. Furthermore, we theoretically establish that single graph convolution using the learned graph in MAGE is mathematically equivalent multiple convolutional steps under conventional graphs. We evaluate MAGE against advanced baselines on multiple real- worldwrldd spatiotemporal datasets, and MAGE achieves competitive performance while maintaining strong computational efficiency. Our code is available at official repository.
Introduction
Spatiotemporal forecasting, a core task in smart city applications, plays critical role in key domains such as energy, meteorology, and transportation [1, 2, 3]. Among the various approaches, graph-based modeling has become the dominant paradigm for capturing spatiotemporal dependencies, where sensors or base stations are represented as nodes and edges encode spatial and temporal relationships. Spatiotemporal graph neural networks (STGNNs) perform message passing over the graph to learn node representations. As result, the accuracy of spatiotemporal dependency modeling in these systems critically depends on the quality of the underlying graph structure.
Early graph learning methods typically relied on predefined priors, such as geographic proximity, to compute pairwise node similarities [4, 6]. However, real-world settings, such prior topological information is often incomplete, noisy, or task-specific [7, 8J. To overcome these limitations, data- driven graph learning methods have emerged more robust and flexible alternatives, enabling
BBnww Wang and Yang Wang are corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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end-to-end learning of latent graph structures directly from data. Notable examples include spatial Transformers and adaptive graph learning methods. While offering greater computational efficiency and flexibility than Transformers, the latter has been widely adopted in spatiotemporal forecasting models such as AGCRN [9), GWNet [7]. and DºSTGNN [10]. These models commonly generate an adjacency matrix A = max (E, ) through two learnable node embeddings E and Eg. Despite these encouraging results, this method incurs a quadratic complexity with respect to the number of nodes, which limits its scalability on large-scale spatiotemporal systems. Moreover, certain seemingly innocuous but persistently used ReLU activation functions degrade the effectiveness of leamingg the underlying graph.
To address these limitations, we propose Mixture of Adaptive Graph Experts (MAGE), novel frame- work that achieves linear computational complexity while offering enhanced expressiveness. First, through extensive theoretical analysis, we reveal that the commonly used ReLUU activation function in existing adaptive graph learning method disproportionately amplifies negative edge weights while suppressing positive ones. This behavior inadvertently reinforces noisy edges during graph learning. thereby impairing the model's ability to accurately capture spatiotemporal dependencies-aa an issue that necessitates removal. Subsequently, we design kernel-based function as approximation scheme for similarity calculation that reduces the computational complexity from quadratic to linear with respect to the number of nodes. However, according to the matrix theory, we show that such an approximation leads to reduction in the rank of the learned adjacency matrix, which in turn limits its representational capacity- so-called low-rank bottleneck. To address this problem, we intro- duce sparse yet balanced cfitture-of-expert strategy, where each expert learns distinet adaptive graph. The sparsity strategy enforces that each node interacts only with compatible experts, while the balancing mechanism encourages uniform activation across all experts, thereby facilitating the learning of diverse graph structures. Finally, we provide a theoretical analysis to demonstrate the strong representational capacity of MAGE capturing adaptive graph structures. MAGE achieves state-of-the-ar performance with strong computational efficiency.
Our contributions are summarized as follows,
Practical Solution, We propose a novel Mixture of Adaptive Graph Experts (MAGE) for efficient spatiotemporal forecasting. MAGE mainly incorporates sparse yet balanced expert assignment mechanism, where multiple experts interact with nodes in a sparse and selective manner, enabling the extraction of expressive and diverse underlying graph structures.
Theoretical insight. We provide a comprehensive theoretical foundation for the design motivation in MAGE, such as insights into edge-level noise amplification in existing methods, low-rank bottleneck, and the equivalence between single-step graph convolution in MAGE and multiple convolutions on conventional graphs.
Empirical Study. Extensive experiments across 17 real-world datasets and 14 advanced baselines show that our method achieves SOTA performance on 94% (48/51) of the metrics while maintaining high computational efficiency and scalability.
2 Related Work
spatiotemporal forecasting is a fundamental task in time series analysis and plays critical role in wide range of real-world applications [11, 12, 13, 14, 15, 16, 17]. In recent years, STGNNs have become the most representative approach for this task [18, 19, 20]. Early STGNNS relied on static graphs constructed from fixed geographic or domain-specific attributes to capture spatial topology [21, 22). However, such predefined structures often fail to model the underlying dynamic spatiotem- poral dependencies among nodes. To address this limitation, more advanced models have been proposed, such as DGCRN [6]. GWNet [7], and D²STGNN [23]. These methods jointly leverage predefined graphs and learnable adaptive graph mechanisms, enabling the model to infer optimal spatial relationships directly from data. Some STGNNs, including AGCRN [9] and MTGNN [24), go even further by completely discarding predefined graphs and relying solely on data-driven adaptive graph structures, thereby achieving strong empirical performance. With the growing popularity of Transformers across various domains, researchers have also developed Transformer-ban based architec- tures for spatiotemporal modeling, such as STAFformer [25] and DºSTGNN [23]. In this work, we focus on adaptive graph learning, a lightweight yet effective paradigm that captures latent node affinities through a simple dot product between node embeddings. Although this approach is simple

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Future Prediction Y
MAGE Block
Mixture -ff-Adapivee Graph Experts
+ Summation
Graph Experts
Linear
Product
L
Feed
K
Block
Norm
Position Encading
Linear
MAGE Layer
Charinel Compress
Norm
Balancer
Top-K
Spatiotemporal Data
Figure 1: The architecture of MAGE for efficient adaptive graph learning.
and widely adopted STGNNs, its computational complexity grows quadratically with the number of nodes. To alleviate this scalability bottleneck, BigST [8] introduced positive random features to reduce the graph construction complexity from quadratic to linear in expectation. Building upon BigST, GSNet [26] models the adaptive graph as a low-rank matrix generated via linear transfor- mations. Other methods [27, 28] attempt to prune the learned adaptive graph during inference to reduce computational overhead. Although these approaches improve efficiency by sacrificing some representational capacity, they generally underperform compared to conventional adaptive graph learning techniques.
3 Preliminary
Spatiotemporal Graph. We use graph G (V, A.) representing spatiotemporal data, where V means the node set with N nodes, & represents the set of edges (c.g.. similarities between nodes) and A E NxN is the weighted adjacency matrix of the graph & that can be generated based on the static method data-driven method. We use 24 E NR/ to represent the observed spatiotemporal graph signal at time step 1, where f indicates the number of feature channels.
Spatiotemporal forecasting, Given the graph and the historical data of the previous time steps X ( xT) I as input, the goal of spatiotemporal forecasting is effectively predict future data Y (2T+I....,
RTpxNX in future Tp time steps output.
Adaptive Graph Learning. Adaptive graph learning typically formulated through reparam- eterization of two learnable node embedding matrices, E,, E2 E Nxd, where de < N is the prescribed dimension of the graph generation embeddings. The adaptive graph is then constructed as [7, 9, 23, 24, 29]:
A = Softmax (ReLU (E, (E, E ERN
(1)
Computing the similarity matrix E, E NxN involves a quadratic time complexity of O (N²dg). which significantly hampers the scalability of the method. Moreover, the ReLU function may amplify negative entries in the latent similarity matrix, potentially leading to the unintended enhancement spurious dependencies (i.e., noise) the adaptive graph A. We develop an analysis of edge noise amplification theory of this phenomenon, which is provided in Appendix A.1. And we also demonstrate the negative impact of ReLU through extensive experiments Appendix D.4.
4 Methodology
In this section, we present the detailed design of the proposed MAGE. As shown in Figure 1, MAGE first employs kernel function reduce the computational complexity of similarity calculation. It then introduces a sparse yet balanced mixture-of-expert method, which adaptively learns the underlying graph topology from the data.
4.1 Linear Adaptive Graph Learning
As shown in the theoretical analysis Appendix A.I, the ReLU function may amplify negative correlations among nodes, potentially leading to the introduction of noise in learned representations.

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Thus, we remove ReLU (-) before Softmax (.) to overcome the edge noise issue. that is,
A Softmax [ E E N
(2)
Thus, the graph convolution of node Us aggregating from Vi,, can be defined as,
H (0) Σ. Sim (E16 E2,) H (c-1) JEV Emev Sim (E E2m) E ER"
(3)
where Ej; means the i-th row of Ej. And H(--)) means the representation of node by in the (c-1)-th
graph convolutional layer. H! means the output representation of node v;. The above calculation process has quadratic complexity with the number of nodes. The aforementioned computation exhibits quadratic complexity with respect the number of nodes. To address this limitation, we
introduce kernel-inspiree approximation approach [30, 31], which approximates the similarity (
matrix via the inner product of two non -eggatvtevve activation functions &
(.), V (-)
H = = 4 (EI,), V (E2;)) H. ((
① (Ei.). (EE))_ (E2,) H. H
(4)
JEV EmEV { (E1;). V (E2m))
( (Ei)) ΣmEV V/ (E2m))
where (:,-) is the vector inner product. And we choose exponential activation with bias c: E14 exp (Ei; + m:), v Egy + exp (E2, + E;) with all ni & Rede satisfying rp Ô and EAA - In (Emev exp (e)))) )). At this point, kernel-based graph convolution can be written as,
da
exp(e
exple
H
(5)
jEV w== exp(e Fm
Emer exp (2)2)
And the adaptive graph convolution can be defined as,
H (c) Softmax (E₂) Softmax E H (c1)) E Nxd
(6)
where first caleulate Softmax E H(c-1) according the law of multiplicative union. In this way, the complexity is O (2 N x d de), that is, linearly with the number of nodes N, because and de are much smaller than N. Detailed derivations of the above part are available in Appendix A.2.
Low Rank Bottlenecks. However, the approximate method often incurs degradation represen- tational capacity. To theoretically characterize this trade-off, we leverage matrix theory, where the rank the learned adjacency matrix can be used as measure of the high-dimensional information preserved in the feature representations. Specifically, the rank of the adaptive graph satisfies:
Rank (A) Rank (Softmax (E,) Softmax (E,)) S min (N, dc) do < N.
Rank (H(*) (0) = Rank R (AH((e)))))) S min {do, N, d) de d.
(8)
Traditional adaptive graph method can yield graphs with full rank: Rank (Softmax E E, N, whereas the kernel-based approximate method achieves a lower effective rank. Consequently, the node representations derived via graph convolution are constrained to a low-rank subspace, limiting their expressiveness and discriminative capability.
N,
4.2 Mixture-of-Exwe -xpert for Boosting Linear Adaptive Graph Learning
We introduce the mixture-of-exper strategy, in which each expert independently generates an adaptive graph. At this time, the adaptive graph convolution with K experts can be expressed
H
(4) A
A
H
as Softmax(E(()) ( Softmax(E5
(K)
TK)T H HERNOO
(9)
Here E [0 denotes the sparse yet balanced weights of K experts satisfying Σk k= 0g = 1. The exact calculation of which will be described in Section 4.2. And E1(A) and E2(*) denote two

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learnable embeddings of the k-th expert, which are used to generate its corresponding adaptive graph. At this point, the rank of node representation generated by the mixture-of -fxpeerr strategy is expressed as.
K
Rank(H(o) S min(d, Σ
min (da, - min{d, Kdo}
(10)
km
When the number of experts K satisfies K 2 [d/dc]. The rank of the node representation matrix will be bounded by d. However, making K too large has little benefit and may even lead to overfitting. as the diversity of features starts to saturate. Therefore, we set K to [d/ dc]. At this point, the multi-expert strategy enhances the rank of the representation matrix, which we empirically validate in Section Experiment 5.6.
To further enhance the representational capacity, incorporate differential mechanism into the adaptive graph learning process. For each expert k, we assign four learnable embeddings: E E²Aa) Eg (k) EAA) E RNxdo, and the adaptive graph is generated in differential manner as follows,
A (k) Softmax(E)" ) Softmax E T) - A Softmax (EE, ( Softmax T
(11)
To maintain the numerical stability of the X, we re-parameterize À as follows,
À = w + exp ((),))))) - e(A3,
(12)
where w E (0, 1) is hyperparameter and λ1 A2, λ3, da E Rd are learnable parameters.
4.2.1 Sparse yet Balanced Mixture-of-Expert
We aim to develop sparse yet balanced mixture-of-experrs system. Spansity ensures that only small and relevant subset of experts is activated for the input of each node, reducing computational cost. Balance ensures equitable activation across all experts over different inputs, preventing over-reliance on any particular subset. To select the desired experts, we first define a candidate pool consisting of Kg > experts, which is denoted as IP = {A 1 Ka 1k=1"
Sparse. For k-th expert candidate, we assign learnable identity vectors 0g Rdd, and then we calculate the affinity between the node representation and each expert:
āa = Sigmoid H (c( of + +76)
1.
7 - +00,
(13)
1 + exp (-7x) exp H (--1) 0
0.
0o.
where normalized 0i means the affinity between node and the k-th expert candidate. The learnable scalar 7h E IR with Sigmoid function is used to encourage the model to generate sharply peaked attention weights, favoring clear preferences of candidate experts.
Balance. Follow the idea in the work [32], we introduce a priority modulator into the expert selection process described above for balanced activation. If the k-th expert candidate is activated more frequently than the average expectation in previous rounds, a negative value 3& is applied to penalize its affinity score. Conversely, if it is under-activated, B& takes a positive value to encourage its selection. Accordingly, the optimal expert selection process becomes:
Cisk + 3k, k arg Top-K (ä,, + Brrr = 1,2,.... Kah. Otherwise.
Osk
(14)
0.
where arg Top-K (.) means the indices corresponding to the Top-K largest values. i.e., for Kg candidate experts, we retain the Top-K experts that exhibit the highest affinity with node vù. Balanced activations are beneficial for learning generalizable semantic graphs.
Finally, we develop load balanced optimization strategy for 35, which computes the difference between the activation count' of k-tth expert N₂ and the average activation expectation across Kg An expert is considered to be activated once if the attention between it and any node is greater than
5

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candidate experts:
Ka
K)
Kg
(N.
K)
Lload
Ns
3x + StopGrad (Nx 3x)
(15)
2
KG
Ka
(N - K) KG
(N.K) K)
Va, Lload
3x + StopGrad (Nr Bp) V
-
(16)
Kg
where StopGrad(() the stop-gradient operator [33], keeping the forward output constant but forcing the gradient to zero. Each of the N nodes selects K affinity experts, resulting in total of N * K expert activations. In this work, we optimize 3g. by symbolic stochastic gradient descent [34, 35] as follows.
Be < Bx p.8gn (V Lloud) = B3 - sg NE (N-K) K) KG
== (1,,,... Ko). (17)
where > 0 is the learning rate of optimization of BR- sgn(-) means the signum function. To promote balanced expert utilization, the model adjusts the activation priority of each expert based on its historical usage. Specifically, if the k-th expert is selected more frequently than the average, its associated parameter 3e decreased, which reduces activation probability in subsequent steps.
4.3
Mixture of Adaptive Graph Experts
The final version of our adaptive graph convolution of one layer named MAGE (.) as follows,
MAGE H diag (a11, O2k. a .ONk) A ( H.
(18)
A (4) ( (k)) (oftmax(E, (K) Softmax Softmax(EE
(19)
where diag (.) means the diagonal matrix. Orsk NxKo denotes the affinity matrix between node v and k - th expert candidate.
4.4 Overall Architecture
We stack L layers of MAGE capture deep-level spatiotemporal dependencies, and the forward process of (-th layer can be denoted as follows,
Z ( FFN, (Norm(H()) + (
(20)
H MAGE (111)) + (1-)))
(21)
where FFN (-) the Feed Forward Network with SwiGLU (+) as the activation function [36]. The input representation ( is the transformation of the input data X combined with spatiotemporal position embedding as follows,
Z O XW, + bo PPRRN
(22)
where Wo and by are learnable parameters, and P E xd is the spatiotemporal position embedding, which incorporates various forms of prior information; further details are provided in Appendix B. The final forecasting is generated as follows:
Ý Z (L) W L+1 + bL+1 (N
(23)
where W L-1 and 1 learnable parameters. Finally, redistribute the dimensions of Y to Tp x N x f for aligning the dimensions.
Experiments
5.1 Experimental Setup
Datasets. use 18 spatiotemporal datasets from four domains: traffic, energy, meteorology,
and mobile communication. Traffic datasets include SD, GBA, GLA and CA in LargeST [37],

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XTraffic [38], PeMS series: PeMSOX (X=3,4.7,8) [39] and PeMS- Bay [5] datasets. Energy datasets include Electricity [40] and UrbanEV [41]. Mobile communication datasets include Beijing Weibo, Shanghai Mobile [42], and Milan Internet [43]. Meteorology datasets include KnowAir [44] and China City Air Quality [45]. Details of these datasets are available in Table 4 of Appendix D.1.
Settings. Our experiments are deployed on the LT platform [37] for all datasets to ensure a fair comparison. All datasets are divided into training, validation, and test sets chronologically in ratio of 6:2:2. We employ three common metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE) to evaluate model performance. All experiments are executed on an NVIDIA A A100 with 40GB memory. The code environment is based on the PyTorch using Python 3.11.5. The length of the input window and forecasting horizon, T and Tp, are set to 12 for all traffic datasets and 24 for other domain datasets. We adopt the Adam [46] optimizer with an L, loss function, a learning rate of 0.02, and predefined milestones decay factor of 0.5. We use only I. = 3 MAGE Blocks with hyper residual connections for all experiments, with a dimensionality of candidate experts in all datasets Kg is set to 16, and the number of activated experts per node K is set to d 4. The learning rate for all [% in the load- ballbaaleeed optimization strategy is 10
128 and graph generation dimension of dG = 32. The maximum number of
dg
Baselines. Our experiments consist of multiple advanced spatiotemporal prediction models, including:
AGCRN [9]. BigST [8], DGCRN [6]. DºSTGNN [23], GSNet [26], GWNet [7], MTGNN [24], PatchSTG [47], RPMixer [48], STAFformer [25], STGCN [4], STID [49], STNorm [50], and
STWave [51].
5.2 Forecasting Performance Comparison
The main results of the forecasting performance comparison are summarized in Table 1. For clarity and readability, we present results on four representative datasets spanning different domains; results on the remaining datasets are provided in Appendix D.2. Methods based on static graph structures, such as STGCN, exhibit limited performance because they cannot capture dynamic spatiotemporal dependencies. GWNet and AGCRN employ adaptive graph learning strategies to improve spatiotemporal modeling. Transformer- based models DD'SsNNN,, STAEformer, and PatchSTG are capable of learning adaptive spatiotemporal patterns directly from data, thereby
Table 1: Performance comparisons. The best and second best mean performance are in corresponding colors. The '-" marker indicates baseline incur out-of-memory issues even on minimum batch size. The Yf' marker indicates baseline is not applicable to this dataset due to the absence of key metadata (e.g.. latitude and longitude). All experimental results are the average of five independent runs.
Methou
SD
GLA
XTralli
MAF
RMSE
MAPE
MAF
RMSR
MMAE
MAP
RMSE
MAPR
MAF
RMSE
MAPR
MAR
RMSR
MAPE
STGGN
19.27
33.57
13.49
23
38.15
17.82
22.22
37.95
14.50
20.68
33.65
15.55
13.55
26,58
31.15
DOCRN
17.79
29.31
12:33
20.55
33.40
16.79
AGCEN
18:38
33.63
13:78
20.04
34.30
16.05
20.26
34.85
12.59
GWNet
18:07
29:97
12:70
20.33
33.37
17.30
20:37
32.65
12.71
19.75
31.71
15.84
15:25
28.55
21.94
MIGNN
18:21
30.99
12.95
21.45
34.91
17.17
21.75
35.35
14.83
1991
32,83
15.11
12.48
23.39
19.50
STNomm
19:36
32.14
12.00 3.955
21.00 1.999
35.28
17.17
21.84
12.99
20.37
33.
15.04
12.00
22.91
18:31
STID
18.05
30.85
12.15
20.65
34.29
16.92
20.40
33.90
12.97
19.04
31.86
14.59
11.62
22,41
19.84
RPMixer
26:01
43.64
18.32
28.5
52.59
26.85
28.55
$1.95
19,00
25.44
47.93
20.54
16.68
43.64
$2.74
BigST
17.08
29:61
11.05
21.15
34.3%
17,80
20.96
34.00
13.
19.32
32.01
14.93
12.13
23.01
21,42
CiSNet
18:75
31.30
12.87
21.3 38
35.3 38
18.04
21:31
34.75
13.
19 60
30,2 14.24
15,50
13.35
24.67
21.09
ST Wave
17.64
29:61
11.83
20.55 56
33.5 5.55
20.22
33.00
12.58
20.67
33.12
15.96
STAFfoomer
19.02
31.78
13.65
31.3 30
34.5 56
13.63
DPSTGNN
12.13
25.60
12.15
21.15
34.09
16.08
Paoch/TG
17.46
30.13
11.24
19:75
33.17
14.05
19:30
33.28
11.38
17:68
29.72
12.86
10:63
20.85
19.41
Ours
16.39
25.04
10.87
19.58
32.79
14.24
18.90
31.58
11.25
17.37
29.37
12.47
10.34
20.45
17.92
Method
Electricity
UrbanEV
KnowAir
China City Air Quality
Beijing Welbo
RMSE
MAPE
MAE
RMSE
MAPE
RMSE
MAPE
MAE
RMSE
MAPE
MAE
RMSE
MAPE
STOCK
2602
2310
14.14
5.91
12.34
19:17
15.77
24.25
57.64
19:56
31.54
25.48
0.5549
1,6851
34,81
DOCRN
2503
2353
18.14
5.22
11:47
18:70
21:11
30.62
65.55 89
21.87
35. 18
35.05
0.5637
1.7842
31.55
AGORN
2115
1847
16.95
5.36
13.30
18.21
16:34
24.81
61.26
19:57
33.65
$1.41
0.395
1.698
53.68
GWNNa
2803
1820
19.4%
5.27
11.57
18.95
15:49
21.85
56.73
18:74
31.73
20:11
0.315
1.677
$1.74
MITGNN
1948
1583
16.53
5.27
11.31
18.40
15.34
24.21
58.70
19662
32.58
30.70
0.3300
16653
$2.50
STNumm
23033
1983
14.92
5.43
11.54
19.24
16.00
24.32
59.46
19.72
33.13
30.04
013721
1.7228
52.15
STID
126.0
1532
12.4%
5.23
11.99
18.24
24.85
61.41
20 54
34.13
30,
03350
1,6730
$2,40
RPMiue
188.
1574
19.19
6.52
13.62
24. 34.80
16.7%
25.05
$4.07
19.05
32.65
78.91
101500
133666
45.58
RigST
1903
1692
19.85
5.49
11.23
19:70
15.68
15
56.53
18.62
11.01
20.37
03351
1.8036
$1.30
GSNN
191.8
1617
14.9%
5.55
11.39
30.26
16.30
24.65
60.57
19:50
33.04
$1.20
033358
1.6763
$3.39
STWine
188.2
1772
11.69
5.04
11.15
17.81
16.35
24.93
61.93
20:26
33.95
33,017
113308
1.6949
$1.38
STAEfomer
280.5
1650
13.75
5.01
11.16
17.64
15:82
24.56
$3.25
19.01
31,57
30,34
118332
1,6810
52.12
DPSTGNN
224.8
2110
17.46
5.00
11.40
46
17.05
15.10
24.51
55.41
1632
33.29
26.10 10
03450
1.7216
31.80
PHelSTIT
/
5.16
11.53
17.89
1608
24.70
56.78
1898
30.17
73
03868
1.7561
32.16
Ours
11.57
4.95
11,00
15.36
53.77
18.52
30,85
36.13
IL7958
1.6477
29.85
7

---

achieving improved performance. However, their high computational complexity hinders scalability. especially on large-scale datasets such as CA and GLA. STID is a linear spatiotemporal modeling architecture that integrates various embedding techniques and achieves performance competitive with GNN-based models. RPMixer captures inter-node relationships through randomly generated projection matrices. DGCRN introduces a dynamic graph that evolves with traffic flow data, but its performance is inconsistent across different scenarios. In contrast, GSNet and BigST adopt enhanced adaptive graph learning mechanisms, achieving both competitive accuracy and good scalability. Our proposed method outperforms all baseline approaches in terms of prediction accuracy, because MAGE enables more comprehensive exploration of the underlying graph topology, thereby enhancing spatiotemporal modeling and leading to superior forecasting performance.
5.3 Ablation Study
In this section, we design following variants of our model to validate the soundness the main component of our model: 'w/o PE" removes all the spatiotemporal position encoding embedding; 'w/o SE' uses only feedforward networks as model backbone without spatial encoder; 'w/o Multi leverages only one adaptive graph expert with K 1; 'w/o LB' reduces the load balanced optimization strategy in MAGE; 'w/o Sparse' sums up all output of alternative graph convolution. Additionally, the combination ablation experimental results for each spatiotemporal position encoding embeddings are in Figure 4 (b) in Appendix D.3.2. As shown in Figure 2(a), the ablation study reveals that 'w/o SE' variant achieves the worst performance. This is because our mixture-of-addppe convolution module plays a crucial role in guiding the model to recognize dynamics spatiotemporal dependencies among nodes. 'w/o PE' variant also suffers from higher forecasting errors, which can be attributed to the fact that the learnable spatiotemporal position encoding can extract piratical and general knowledge during training. The performance deration of both 'w/o LB' and "w/o Sparse' variants indicate that sparse and balanced graph convolution possess better performance than dense graph convolution and graph convolution without balancing loading, respectively.
graph
The total number adoptive graph number
17.0
CCND Walee
50
2
55
MAR
wis PE
MAFE (1))
IS
Saaa Erihady Electriiiy Ntralle
15.5
14
14
18
70
24
10
1675
wes Malti
The number of
wis Spane
Oars
MAD
11.25
16.0
153
1675
10
21
12
14
15
15
(a) Ablation study on all datasrts.
(b) Hyperparameter Sensitivity Experimets on SD dataset.
Figure 2: (a) Ablation study on all datasets. (b) Hyperparameters sensitivity experiments on KG and K in SD dataset. The yellow star marks the optimal hyperparameters.
5.4 Hyperparameter Sensitivity Experiment
We analyze the sensitivity of MAGE to its two key hyperparameters: the total number of expert graphs KG and the number of activated experts per node K. Using the best-performing configuration as the baseline, we vary one parameter at time while keeping the others fixed, as shown in Figure 2(b). We report MAE and MAPE on the SD dataset for evaluation. The total number of candidate experts varied from 2 to 32, and the number of activated experts per node ranges from to 16. When the number of candidate experts KG is too large, the model struggles to select the most suitable ones, leading to suboptimal performance. Moreover, activating too many experts each time introduces redundancy and degrades prediction accuracy.
8

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5.5 Efficiency Comparison with SOTA STGNNs
As shown in Table 2, our model achieves the highest prediction accuracy among advanced STGNNs while simultaneously demonstrating the lowest computational complexity and the highest efficiency. STWave combines a naive graph convolutional network with decomposition techniques. Although exhibits relatively high efficiency among the baselines, its performance is only modest. STAFformer employs a standard Transformer architecture with quadratic complexity in the number of nodes, resulting in low efficiency. DºSTGNN integrates multiple dynamic graph convolutions, which further increases complexity and leads to training speeds that are more than 118 times slower. On the GBA dataset, it is 960 times slower than our method. In comparison with another advanced model, PatchSTG, MAGE achieves up to a 4.7 times speedup in inference and reduces memory consumption by up to 1.72 times. Moreover, by avoiding complex Transformer architectures that introduce large number of parameters and high computational costs, our model substantially lowers memory overhead. It requires up to ten times less memory than D'STGNN and STAEformer.
Table 2: Efficiency comparison with SOTA STGNNs. Memory: The maximum memory usage (MB) during training. BS: The maximum allowable batch size during training (up to 64). Train: Average Training Speed (s/epoch). indicates the relative percentage increasing regarding MAGE.
Method
(16)
GBA (2352)
UrbanEV
MAE
Memory
BS
Train
MAF
Memory
B8
Train
MAE
Memory
BS
Train
STAFfermer
19.02-15.754
36 41.7516 384 2045% 21,
fu.30 30,518 290 oms
2529
4330 sns
5.09.,.115
53,6800
4100 745 neceol
STWine
17.44
26,524 nou
64
411
20.5 56
m. 40.564
200.05
26 sssass
1034,
4794
5.04 1.32
36362
18
200
D' D'STGINN
17.13-
8.38
31
11
442
1999
21.13
now
90,102
5527
0999 ans
5.12
90006
2257
PitchSTG
17.46
7,12
64
101
19.75
27,852
64
326
5,36:
12,106
64
25
Durs
16.99
3,662
19.58
10,230
4.95
5594
5.6 Low Rank Bottleneck of Various Models using Adaptive Graph Leaning
In this section, we expose the rank bottleneck in existing adaptive graph learning methods. We compare our model with representative approaches- approaches-D'STOne D'SSGNN, BigST, and GSNet-by measuring the effective rank of node representations after graph convolution, computed via SVD with a threshold of 10-8, To ensure fair comparison across models with varying embedding dimensions, we normalize the rank by its theoretical maximum. As shown in Figure 3 (b), STGGN achieves the highest normalized rank (60%), reflecting its strong representational capacity due to standard adaptive graph learning. In contrast, BigST and GSNet adopt linear approximations to reduce computational complexity, resulting in a significant drop in rank (retaining only 20-40% of the theoretical upper bound), which indicates a notable loss of expressive power. Under linear computational complexity, MAGE attains 80% of the theoretical rank limit, outperforming all compared efficient variants. This gain is attributed to its multi-expert adaptive graph mechanism, which supports more diverse and informative spatiotemporal modeling without sacrificing efficiency.
LaggST-SD
Electricity
Wosman Loage Mati (Ss)
LB Usage Bado (h)
wir
42
Staak
Warted Expects Activated
Load Balencing Usoge Ratio: 1/16-6, 25%
MAGE
Righ
Gshe PPo0u
MADZ:
RighT
Othie DPSicoo
Craph Eisperte
(a) Rank ratio (%) comparison
(b) Sorting Usage Ratio (% Graph Experts on SD
Figure 3: (a) Rank ratio (%) comparison on the outputs of adaptive graph convolution which is the ratio of the true rank of the output of the adaptive graph convolution to its rank upper deterministic boundary. (b) Usage Ratio (%) of all graph experts on SD dataset. The expert order is sorting by the usage in "w/o LB" variant.
5.7 Evaluation of the Balancing Strategy in the Mixture-of-Expe System
We evaluate the balancing strategy by analyzing expert utilization on the SD dataset. As shown in Figure 3(b), we compare the full model with a variant that removes the load-balancing component
9

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(w/o LB). For both models, we first extract the affinity scores between nodes and experts, and then count the activation frequency of each of the 16 experts. Our model consists of 16 experts, each activated approximately 6.25% (1/16) of the time, indicating a well-balanced usage across experts. In contrast, the w/o L.B variant exhibits highly skewed activation patterns, where certain experts are heavily favored while others are underutilized. This imbalance limits the model's ability to capture diverse spatiotemporal patterns. The balancing mechanism in MAGE ensures more uniform distribution of expert activations, enabling the learning of richer and more diverse adaptive graph representations, ultimately leading to improved performance.
5.8 Pareto-Optimal Trade-off Study Between Linear and Full-Rank Adaptive Graphs
To further examine the seeeiency-pefformance trade-off within MAGE, we conduct a systematic study on the proportion of linear kernel adaptive graphs (Eq. 11) versus naive full-rank adaptive graphs (Eq. 2) without ReLU. Specifically, we fix total number of graph experts 16-consistent with the optimal configuration in the main experiments-ant progressively adjust the mixing ratio between the two graph types, while maintaining the original balanced and sparse expert activation constraints.
Table 3: Pareto-optimal study of performance-<ffccinncy trade-off's of adaptive graph type.
Linear: Full
MAE
RMSE
MAPE
Memory
Training
Naïve
0:16
16.52
28.35
10.91
4,308
46 s/epuch
4:12
16.53
28.29
11.01
3,998 MB
35 s/epoch
8:8
17.10
28.75
11.31
3,860 MB
30 s/epuch
12:4
16.29
28.22
10.88
3,696
24 s/epoch
Ours
16:0
16.29
28.04
10.87
3,662 MB
22 s/epoch
As shown in Table 3, with the proportion of full-rank adaptive graphs increasing, memory consumption and training time rise substantially, yet without yielding any noticeable performance improvement. In contrast, the pure linear configuration ("Linear:Full" 16:0, the default setting in MAGE) achieves comparable or even superior forecasting accuracy with minimal resource overhead, indicating that the original MAGE design already lies Pareto-optimal point in the accuracy and computational efficiency trades-off. Therefore, our linear adaptive graph convolution achieves high predictive performance while maintaining excellent computational efficiency. We further extend the above experiments to investigate the model's inherent preference between linear and full-rank adaptive graph convolutions. The results reveal clear tendency for the model favor our proposed linear adaptive graph formulation. Detailed experimental settings, analyses, and results of this study are provided in Appendix D.5.
6 Conclusions
In this paper, we propose MAGE, a novel and efficient framework for adaptive graph learning with linear computational complexity. MAGE combines kernel-based approximation with sparse yet balanced multi-expert architecture. The sparsity mechanism ensures that each node activates only the most relevant experts, while the balancing strategy promotes uniform expert utilization across the network, leading to more robust and representative graph learning. We further provide theoretical insights into the edge noise issue present in existing adaptive graph learning methods. Extensive experiments across multiple spatiotemporal datasets from four distinct domains consistently show that MAGE outperforms state-of-the-arr baselines while maintaining excellent computational efficiency.
Acknowledgment
This paper is partially supported by the National Natural Science Foundation of China ANo.1222790) The AI-driven experiments, simulations and model training were performed on the robotic Al-Scientist platform of Chinese Academy of Sciences.
10

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[78] C. Qian, A. Manolache, C. Morris, and M. Niepert, "Probabilistic graph rewiring via virtual nodes," " Advances in Neural Information Processing Systems, vol. 37, pp. 28359-28392 2024.
15

---

A Theoretical Justifications
This section presents the theoretical proofs. In the first subsection, we provide detailed mathematical proof to support our significant observation: ReLU introduces edge noise in adaptive graph topology generation. the second subsection, we mathematically demonstrate the effectiveness of a single convolution on the adaptive adjacency graph convolution, which we refer to as 'You Only Convolve Once' (YOCO). Furthermore, we prove that our adaptive adjacency matrix generation method genuinely achieves YOCO. We pack all the Theorems in the red color box, all the Lemmas in the cyan color box.
A.1 Magnify Noisy Edge by ReLU Function in Adaptive Graph Learning Method.
Theorem 1. Edge Noise Amplification Theory
Let E | 1e Nxde be the graph generating embeddings where all elements belonging to them satisfy an independent normal distribution N' (0, o²) with a > 0. N corresponds the number of nodes and dG N is the given dimensionality of graph generation embeddings. The Adaptive Graph with or without ReLU (-) are respectively calculated follows,
A Softmax (ReLU EE, E RNNN, A Softmax (E,E,) RNNN (24)
Then, the calculation of Adaptive Graph A R will lead to more edge noises than A. Specifi- cally, there exits, (1) If nodes and j have positive similarity, then A S A.;
(2) If nodes i and j have negative similarity, then with high possibility A 2 Aij.
Proof: (I) We first consider that the nodes i E V have no negative similarity Sij Ede (1) (2)
ik Gjk >
O, and let Ω {L Vlsa 2 0} S V be the set of nodes with negative similarity to node i, ReLU (sty) = max {stg. 0), then
exp (8t)), j Ωi 11,
exp (ReLU (sir))
(25)
j 2
and the final edge weights from node j to node under two calculation satisfy,
A R = exp (RcLU (Siz))
cxp (sig) Eien, exp (ReLU (sæ)) Σien, exp (sa) + (N - ((2))
(26)
exp (St
exp (81z)
Eien, exp (sa) Emgo, exp (Sim) Eien, exp (su) Aij-
If 31" E - Ω2 such that sg. < 0, i.e. [2a < N. then the above inequality A. < Aij is strictly valid.
(II) Then we consider that the nodes i,j E V have negative similarity, i.e. 81j = = Sik Gj
By Lemma 1 (1), find A, 2 Aij when $i (-20, In pty] where Pij is the ratio of the sum of the edge weights from nodes other than j to not containing ReLU (-) calculations to the sum of the edge weights from nodes other than j to containing ReLU (+) calculations as follows,
ΣiEr, exp (sa) + EmpRi M5) exp (sim)
Pij Eien, exp (su) + (N 14yl 1)
E (0. 1].
(27)
Although we still find A Aij when sij (In Pij. 0) by Lemma 1 (2), we make the expectation [py] asymptotic for shorting the high possible length of the interval (In (Inj 0)
approaches 0 by controlling a suitable and reasonable a, such as a = of V(N - -1) -1//d) through Lemma 2. Hence, the probability that s,y falls within the interval (In Pij. can be controlled to be very low, that with high probability there is stj E (-o0, In pty] and A > Aij.
16

---

Lemma 1.
If ė and j have negative similarity, then
(1) A 2 Asy when Sty (-00, piyl:
(2) A Asy when Siz E (In Ptj. 0),
where Pij is the ratio of the sum of the edge weights from nodes other than j to not
containing ReLU (.) calculations to the sum of the edge weights from nodes other than j to containing ReLU (-) calculations as follows.
Pv = Eien, exp (sa) + Emggi, \ii exp (sim)
Eien, exp (su) + (N Payl - 1) E (0.1].
(28)
Proof. (I) If si E (-o0, pıj), i.., sig I In Pig. then
EiEn, exp (su) + E Emer. (1) exp (5im)
(s1)) $ Pis Eien, exp (sa) + (N [Gas] - 1)
(29)
Σ JER
exp (8it) + (N - 1041 - 1) exp (8iz) -
Σ
exp (sit)
exp (8im)
(30)
mg Mst
exp (st) + (N - ((21)) exp (81)
Σ
exp (su) +
Σ
exp (sim),
(31)
JER
lEf;
mER
exp (81z)
(32)
Lien, exp (su) Emer, exp (8im)
Eien, exp (8u) + (N [52,]
exp (ReLU (8iz))
Aij S EieR, exp (ReLU (sa)) + Empr, exp (ReLU (sam))
(33)
Aj
(34)
(II) If siy E (In Pij, 0), that is sij In Pij, then just reverse the inequality sign from Eq. 29 to 34 can complete the proof.
Lemma 2. The lower bound of the expectation of negative similarity.
If i and have negative similarity, then the expectation EE [piy] satisfies
1 doo (1 } doo4 +1, (o=o(V(N-11)/) -1) -0). (35)
E [pyl 1
2\
(N-1)
In fact, the above inequality can be further relaxed
1 E Jayl >1 V
doo4 4 (1 - doo
>1 2
+ 1-1, (N - +00) (36)
-N-
but this relaxed o-independent lower bound is also extremely close to I for hundreds to tens of thousands of values N' taken in practice.
Proof. Since all elements E,, E, satisfy an independent normal distribution A (0, o?). then the expectation of similarity between node i and j is,
de
de
E [tyy = E
(1)
(2)
E
E
Cik
E
(2) Cjk
Σ
0.
(37)
k=1
(1) c and (2)) are independent for vk=1,2,...
17

---

and still since (1)
isk and Ej are independent of each other for arbitrary & 1, ... dc. the expectation and variance of similarity between node i and j is,
(2)
de
de
V [1,s]]
V/
(1) il
(2)
jk
jk
[c
W 5
+
IE
ck
jk
+ V
Sik
E
(38)
(1)
(2)
(2)
+ V ( 0)
W
v
5
+
de
de
doo
ak
1=
Then estimate the expectation and variance of edge weights exp (Siz) in Softmax (.) operation before normalization by an approximation methods based on Taylor series expansions [52] at the expectation IE [Sig] of Sij. Concretely,
to0
exp
(E (Sul)
(8ij
E 1869))
exp (8ij)
(855 E (815))
(39)
r!
r!
+00
E [Naj - E [ss]))]]
(sts - E [st]])
E (exp (sij)]
Re 21+ I
E
[ssy]] +
V [8ag]]
1+0+
1
2 doo
(40)
2
It is important to note that when we take = old W [syy] can be arbitrarily small by choice, so the effect of higher order terms can be ignored and the above approximation will be more accurate.
The equivalence definition of variance [exp (s:g)] E (exp (2s,,)] (E [exp (ssz)))² hence we use same approximation methods to calculate E [exp (28:z) as follows,
2" exp (E (2sig]) r! (sig [str])
2" (8ij E [sg)"
exp (2s:z)
(41)
r!
+00
2 [(si]
EE [s
E (exp (28tj))
RJ 1 +1 [soy - E [s,]]] + 2E (
EE (s1s)))]]
r!
1 0 2V [siy] = 1 +2
(42)
Then the variance of exp (8iz) is,
V lexp (sig) =E (exp (2stj)] - (E (exp (st))))² 1 ~ (1 +ddoo)
1 - 1+ 2 _doot)
(43)
1 doo doo (1- (-1dooo)
1
=do
(44)
Let Gi = - exp (su) + Emegr exp (8im) and Gh Eien, exp (su) + (N - |12.| 1),
then the definition of Pij is,
Gij Pij
(0. 1].
(45)
RR
18

---

Then we can get that,
E Texp (su) (NV (1 (1022000)))
[Gy
(46)
[
V [Go]
V [exp (s=)) = (N 1)o (1- 4 ddoot)
1
(47)
E
[<
R
E exp (sa)) + (N [12g] - 1) = R2ll + 2 ddoot)
+ - (N-11/ -1)
(48)
lER;
V [
Σ
Vexp (sa) = 12gldoo (1- 1
(1-2 doot)
(49)
lE0
The expectation of Pij can be expressed as,
E [ppss]] = E [S5 = E [Sg 1/53]] = [(y) E [1/4]] + Cov (5.. 1/5.))
= E [Gig] E [1/C]]
+ Corr (G.,1//33 [Ga V [1/ç]]
(V
(50)
2 E [iy
E [//
[
Where Cov (Sig, 1/53)) is the covariance between Siy and 1/C.j. and Corr (Sy,, 1/5,)) the correlation coefficient between Çij and 1/C.. We compute E [1/C[] and V [1/C]] by a similar Taylor expansion approximation as above.
{[-1,
(-1)"
1/CB
E [cB]] r+I (5 - - E
(51)
E
[1/C9]]
[C
[ r+1
IE
(c# - IE [<g1))
E
[(C
E
[<:11
E [cH]]
E [c/]]]
W |55
E [<:[]
E [c:5]
V [
IE [<:]]
[c 2
(1+ (1+r7y)
(52)
E [c
19

---

where Ti. V [<:3] [<:1]
Similarity there is V [1/5,] E [1/C3)) (E [1/<5])* and
the {1//5)) |
can be calculated as,
(r+1)
(1/58)² Σ
[4R] r+2 (c E. [c E
(53)
E
(1/53)*
1)"
+++1
(5 E [<311*]
[ r+
3E (< E [<:]] 2
2 [(5 - R [<5]]]
E [2/
E3
[c,]]
3V [<,]]]
E [c/]]
E [<:]]
V [s:5]
1 3
(54)
E [<3]]
|<:33 2
(1 + 3T,z)
(55)
E [</]]
Then the variance of 1/4, can be expressed as,
v [1/c,]] = E (1/c/)) (E (A/C/]) (1+ 3TTy) - (1 + +T²)
-
Tay (1 T(t)
(56)
[11]
IE [4]]]
12
Claim.1 in Lemma 2: EE [Gg]] [1/C/]] 1. fact,
1
[c
E [G [1/C:]] (N 1) [+ doo
1
[[cH]]
m 2
E
2 (N-1) - 1) 1 5 doo
E R
(N 1) (1 + 1d
[2al (1 + 1 (N 12 -1) 1)
(57)
(N - 1)
12at+ +
N W-112111
1+3da
(N 1)
[2 +
N --
(N 1)
= 1.
(N - 1)
The first inequality in above Eq. 57 holds since V [s: |9,1 (1 2dco")
and IE [s]]]] 0.
20

---

Claim.2 in Lemma 2: V [G:)] v [1/4/]] < doo
4(N-1) In fact,
V = (N - 1) doo 1 doot)
Taj (1 - Tap)
4
E [S:]]]
(N -W-11 1w-1) 11-1d doo (1 - Hdoo*)
Tij (1 T(p)
[|a.]] (1 _doo)) + (N - -|52p| - -1]]
(N -1) (1 { )doo*) Tij (1 1-TY T(p)
(Qil($do0) + (N - 1)]²
(58)
(N 1) doo (1 ddoo
(N - 112
Aoo 1 (1 ddo0
4(N 1N)
1
(N -)
The first inequality holds since the |[2,] in the denominator is no less than 0, and the maximum tight value-independent upper bound of formula Tij (1 Tig) is 1/4. The second inequality holds since the maximum tight value-independent upper bound of formula doo (1 2d is
dao 2daot)
By above Claim.1 and 2, we have E (ea E [1/5 > 1 and Vis [1/c5) < Finally obtain
4(N-1)
doo (1- 3doo4) (N-
B B
(59)
A.2 Scalability Graph Reparametrization Generation Enpowered by Kernel-like Method
We focus only on operations with the convolution procedure on adaptive graph, i.e.,
A kg H Softmax EE,E,) H()
(60)
We cancel the calculation ReLU (-) before Softmax (-) to overcome the edge noises issue, that is,
A = SSottma (E, E. E NNN
(61)
The calculation complexity of Eq. 61 is still O N²) since the non-linear operation Softmax (-) forbids the law union for multiplication among three matrices E,, E, and H(),, and the similarity matrix E, E, E.J N need to be counted and cost O (dg complexity. Then we introduce a kind of kernel-like method to linearizable simplification for adaptive matrix. A one layer adaptive graph convolution without parameters can be expressed as follows,
H (++1) A kg H (1) (H
(62)
We consider the representation of node E V after adaptive graph convolution, i.e., the i-th of A *g H as follows,
exp
do km
(1) (2)
Ejk
H
(
AH
Σ
Rg
(63)
Emey exp
JEV
kiml eik mm
Hence the adaptive graph convolution is actually equivalent a weighted average of the spatiotem- poral representation of nodes with weights exp (+-) is the vector inner product, which is the exponential activation of the inner product of two graph
[Ede (1) (2) exp ((e))) (1) (2)) O where
21

---

generating embeddings E1. Eg corresponding to different nodes. We can define an universe form of adaptive graph convolution as follows.
Sin(e, (1) (2) (1)
AH
[0]
ER
(64)
jEV EmEr Sim(e) (1) emm)
where the binary function Sim (.-) : Rdc x Rd + R U (0) is positive-definite kernel, com- puting the similarity between two embeddings as the weights. Take the example in Eq. 63 above, i... Sim(e, (1) ) = exp((e,")
( , e the exponential activation after inner product computation guarantees the non- noggtiigtyi of similarity, but it also restricts the implementation of the multiplicative union law, leading to the introduction of high complexity. We therefore draw on the kernel-like method [30, 31] to ensure non-negativity of similarity by introducing a non-negative activation function before the inner product computation. The objective to eliminate the activation opera- tion following the inner product calculation, whilst preserving the non-negativity of the similarity calculation. When all elements of the object of inner product satisfy non-negativity, the result of the inner product is naturally non-negative. To this end, two non-negative activation functions & (-), v (.) : R + R2+ U {0} as kernel-function are incorporated prior to the inner product, thereby ensuring that all elements of the inner product object satisfy non-negativity.
Sim(e (2))
( (o(e(t). ((e))))
(65)
Thus by using the above kernel-like method we can rewrite the adaptive graph convolution in the form as follows:
(o(e,")) (((e))).((e))))) H
()) H (1))))
A kg --=
M- jEV Emer (D(e ((el)))
(o(e!")) ).E V(em (2)
JEV
(66)
(o(c(")) ExEev (2)
H (((e ), (. Emev ((em) (2)
R
We are therefore able to use the law of multiplicative union to prioritize the inner product of (ee))) (2))
and H (1) Here we choose differentiable weighted nonlinear functions & e! exp(e," (1) +m,, V (2) + exple,") + Eg) with all m7 E E Rg then
We are therefore able to use the law of multiplicative union to prioritize the inner product of ((e,))) ( and " Here we choose differentiable weighted nonlinear functions & e + exp(e." (1) + n:), V
o + exp(e)") (2) + Ei) with all m)s Eg E IR:0o then
exp e + n1:) Ejev exp e,
(1)
(2) +Ei) H.
A Kg H()) [1]
exp
( + +m:)
EmEr exp em + +Em
k= exp
do
Cik + Takk)
(EjEV exp
(0) + +G,) H!()) (1)
(2)
[k]]
Ede, exp
(1)
+ now Emer exp
Cmi + Emo
(67)
da Σ
k m= exp Eik + Mikk
Ejevexp exp
EjK + Esit H
(2)
Emer do
exp
c + ema + 1ew + Emar
(1)
(2)
Ede exp
(1)
(2)
+ mak + Eg H!!) (1)
+ Ejk
JEV Emey Id exp G + emw + Thw + Emw
(1) (2)
Hence this kind of adaptive graph convolution is actually equivalent to l weighted average of the spatiotemporal representation of nodes with weights Ede, exp (e + (2)
ejk + thk + Eys > 0.
22

---

Once we know how the elements are computed in Eq. 67, we wish to rewrite the equation into the general matrix computation like Eq. 60. as follows.
Dde des km
exp Ei + jk Tjk + Egs H
(1)
(2)
(A* kg H()) [8]
Σ
jEV
Emey de
cxp
Ci + (2)
(1)
+ 7io + Emw
(68)
exp
sk + Vlik
(2)
exp
jj
(2)
+ Eyit
H (1)
jEV km
m=
exp Ei + 1)w
(1)
Emeyex exp
Cmmw + Smen
Let Eyk- In (E Eee exp (e (2) )) i.e., exp exp (Emev exp (ck))) then
de
exp Cik + 17sk
(1)
exp
Ejk
(2)
Emev exp (c
[1] = A H
JEV k=
do
w=1
exp
Giw
(1)
+ Niw
E Emer exp
(2) Cmme
EmeD exp (emm)
dG
exp
Eii
(1)
Tlik
exp (c
(2)
H
(69)
JEV
da
moml
exp
Cin + Niw
EmEr exp (mm))
If 77: 0, then the above Eq. 69 implies that,
A G H Softmax (E,) Softmax (E,) H
(70)
This approach not only facilitates the calculation of similarity but also ensures the implementation of the multiplicative union law without activation after the inner product, thereby reducing the complexity to O Nd (dg < N) about linear complexity of nodes number N.
A.3 Linear Combinations of Matrices to Raise Rank
Theorem 2. The rank upper bound raising property of matrix addition.
For any matrices M1, M MR
Dix their exists,
Rank
Σ
MR
Rank (M²)
(71)
Proof. We can prove the theorem by means of chunked matrices. In fact,
M
M2
Rank (M;) Rank
M
M
Mi + Ma
K
Me
M
2k M
(72)
Rank
M
K
2 Rann
Σ M
k=
23

---

B Spatiotemporal Position Encoding
Spatiotemporal position encoding aims to distinguish data points by assigning them learnable em- beddings that encode spatial and temporal positions, typically through concatenation [53]. Building on prior work [49, 54], we employ spatiotemporal embedding techniques to incorporate informative priors such as time-of-day and teyyoof-eek. Integrating these meaningful representations into the model enhances its learning capability through effective positional prompting.
Concretely, we utilize three learnable positional encoding embeddings: spatial embedding Ps E RNxd timestep of day embedding Pr E RRd and day-of-week embedding Pn Rd. Spatial embedding Ps assigns a learnable embedding to each node to dynamically capture the spatial property of nodes. The timestep-of-day embedding Pr IR and day-of-weck embedding P R" allocate corresponding learnable embeddings to each time step, dynamically extracting the periodicity of the spatiotemporal data. We use clement-wise addition form our spatiotemporal position encoding P as follows,
P= Ps Pr + PD E E
(73)
Unlike existing approaches that rely on concatenation, our addition- based method is theoretically equivalent to concatenation in terms of representation capacity, but offers greater computational efficiency. This approach serves two key purposes: (I) it avoids increasing the dimensionality intermediate hidden states, and (2) it potentially reduces the number of hyperparameters associated with the positional embeddings used in concatenation-bare methods. The theoretical justification for this equivalence is provided in Appendix B.I.
B.1 Equivalence Between Addition and Concatenating for Spatiotemporal Position Encoding
Theorem 3. Equivalence between + and for Spatiotemporal Position Encoding
Let X be the input data, P's E R Nxds P E Nxdr and PbD E Nxdo are spatiotemporal position encoding for concatenation with weight parameters Wi e R(d+ds then their exits Ps, Pr, Pn E R.Wxd and Wo e Rdxd such that,
X P W. = Xw, + Ps + Pr PP P
(74)
Proof. In fact, the weight parameter W/ be viewed
W [W] w W [W] IR (dddtd
(75)
where Wo Rdxd W E Rddxdd WT R Rddrx
WJ E Rdo
xd are the composition of the first
dimension of Wy Then their exists,
P P PP, |P' P/ Pb) W. W / II
(76)
-XW, + P W + P/ Wr+ P.WD. Then let Ps P'S W. s, Py P Wr PD P W D. and we have,
P W = XW, + Ps + Pr + Pp Nxd
(77)
C
Related Work
Deep learning in time series analysis. Deep learning has shaped rich and diverse ecosystem for a wide range of time series tasks, such as forecasting [55, 56, 57, 58, 59], classification [60], imputation [61], and anomaly detection [62, 63, 64]. In recent years, neural architectures tailored for temporal data have advanced rapidly. Notably, MLP-based models [65, 66, 67] have emerged as highly efficient and scalable solutions, offering lightweight yet competitive performance. Meanwhile, Transformer- based methods [68, 69] lead in modeling power and predictive accuracy. Alongside architectural innovation, there has been growing interest in optimization strategies specifically adapted
24

---

to time series, aimed at improving training stability and robustness [70. 71, 72]. Furthermore, an increasing body of work focuses on downstream applications, seeking align time series modeling with practical, domain-driven goals 74].
Within this broader landscape, spatiotemporal forecasting represents a specialized branch of time series prediction. In contrast to generic forecasting tasks that model only temporal dynamics, spatiotemporal prediction focuses on short-term behavior shaped by structured spatial relationships, where nodes or sensors display strong, learnable interdependenniis- such as in traffic flow, air quality monitoring, or weather systems [75, 76]. The core challenge in this area is to jointly model temporal evolution and spatial coupling in dynamic environments, a problem that continues to drive innovations in model architecture and computational efficiency.
Load balanced optimization strategy. Previous work such as DeepSeek [32] also follows a similar principle by applying the concept of balance to the design of large language models, demonstrating that balanced expert allocation can enhance model performance. However, their study focuses primarily on language modeling tasks, which differ substantially from our setting, In contrast, our work targets traffic forecasting-4 distinct spatiotemporal prediction problem. Beyond adapting this idea, we make two key contributions. First, we provide theoretical analysis of the balancing optimization strategy, including formal derivations that strengthen the interpretability and principled foundation of the approach. Second, rather than directly transferring the balancing concept, we ground it in matrix rank theory to formally justify its effectiveness in adaptive graph learning. This theoretical formulation not only clarifies why balancing improves performance but also enhances the overall interpretability and rigor of the palance-based paradigm in our context.
D
Experiments
D.1 Dataset Description
The description of used spatiotemporal datasets are shown in Table 4.
Traffic Domain. PeMSOX datasets (where X = 3, 4, 7, 8) and PeMS-Bay are provided by the PeMS (Performance Measurement System) operated by the California Department of Transportation (Caltrans). These datasets with general-scale record traffic sensor data from multiple highway regions across California, with a sampling frequency of 5 minutes. The four larger-scale datasets SD, GBA, GLA and CA collectively referred to LTrgeST, are also sourced from the PeMS system. The temporal resolution of these datasets is aggregated to 15 minutes in our experiments and we only choose the year 2019 in experimental comparison corresponding range in scale from 716 to 8,600 sensor nodes. XTraffic represents an even larger spatiotemporal system, comprising 16.972 nodes. Some of the above traffic dataset has capturing traffic flow, speed, and occupancy. More details are in Table 4.
current works [37], and they
Energy. Electricity dataset is widely used benchmark for multivariate time series forecasting tasks. It records the hourly electricity consumption of 321 users or regions from 2012 to 2014, with a temporal gramularity and sampling freeuency of one hour. UrbanEV is a real-world dataset collected from 18,061 public charging stations in Shenzhen over a one-month period (from September 1, 2022 to August 31, 2023). The data is aggregated into 1,682 spatial regions. Temporally, the dataset has time granularity of 5 minutes, resulting in a total of 8,640 time steps. Spatially, it covers 247 traffic analysis zones (nodes), forming a structured graph representation of urban electric vehicle charging demand.
Meterology. Chinese Cities Air Quality (CCAQ) dataset comprises AQI data and corresponding meteorological attributes from 209 cities in China mainland, spanning twenty-eight months (January 1, 2016, April 30, 2019) with hourly temporal resolution. For our quality forecasting model, we still focus on PM2.5 as main prediction object. KnowAir dataset comprises PM2.5 measurements and corresponding meteorological attributes from 184 cities in China mainland, spanning four years (January 1, 2015, to December 31, 2018) with three hour granularity.
Mobility. Beijing Weibo dataset contains blog check- data received from 528 regions in Beijing through the Weibo application from January to December 2023. The Weibo application is a main- stream social media platform in China, with 590 million monthly active users as of 2024, offering extensive coverage. The data points are aggregated at one hour intervals. Shanghai Mobile dataset [42] comprises over 7.2 million call records generated by 9,481 mobile phones accessing the internet
25

---

Table 4: Description the Spatiotemporal Datasets in the Experiments. M: Million (10%). B: Billion
(10%). Data points is the multiplication of nodes and the total time steps.
Domain
Datasets
Nodes
Edges
# Features
Time period
Frequency
Data Points
PeMS03
358
546
3
09/01/2018 ~ 11/30/2018
mins
9.38 M
PoMS04
307
338
5
01/01/2018 ~ 02/28/2018
mins
5.22 M
PcMS07
883
865
3
05/01/2017 ~ 08/06/2017
mins
24.92 M
PeMS08
170
276
5
07/01/2016 - 08/31/2016
mins
3.04
PcMS-Bay
325
2,369
3
01/01/2017 ~ 06/30/2017
mins
16.94
Traflic
LDD
716
17,319
3
01/01/2017 ~ 12/31/2021
mins
0.38 B
LBB
2,352
61,246
3
01/01/2017 ~ 12/31/2021
mins
1.24
LargeST-GLA
3,834
98,703
3
01/01/2017 12/31/2021
mins
2.02 B
LargeST-CA
8,600
201,363
3
01/01/2017 ~ 12/31/2021
mins
4.52 B
XTrallic
16,972
870,100
3
01/01/2023 ~ 12/31/2023
mins
1.78 B
Electricity
321
101,323
5
01/01/2012 ~ 12/31/2014
hours
8.44 M
Energy
UrbunEV
1,682
1,989,848
5
09/01/2022 ~ 08/31/2023
hours
14.73
KnowAir
184
3,796
01/01/2015 ~ 12/31/2018
hours
2.15
Meterology
China City Air Qualtity
209
4,321
01/01/2017 04/29/2019
hours
4.26
Beijing Weiba
528
244,942
3
01/01/2021 ~ 01/01/2022
hours
55.50
Mobility
Shanghai Mobile
3,042
9,090,300
05/31/2014 11/30/2014
hours
13.36 M
Milan Internet
10,000
52,743,034
3
11/01/2013 ~ 12/26/2013
hours
43,93
via 3,233 base stations from June 2014 to November 2014. The data time interval is also one hour. Milan Internet includes multiple mobile traffic features: outgoing calls (CALLOut), incoming calls (CALLIn), sent text messages (SMSOut), and received text messages (SMSIn). These features encompass mobility records collected over two months, from November 2013, to January 1, 2014, across 400 regions. The data time interval is set to hour. We use the Internet subdataset for fair performance comparison.
D.2 Experiment Analysis
We compare the performance of MAGE and SOTA spatiotemporal baselines common PeMS series datasets: PeMSOX (X=3,4.7,8) and PeMS-Bas Bay.. As shown Table and Table 6, MAGE basically dominates the optimal performance due the powerful dynamic characterization capability of the multi-of-addapiive-naa graph module. MAGE is able to capture more accurate spatiotemporal dynamic pattern by dynamically selecting multiple efficient adaptive graph convolution results. However, the performance of adaptive graph convolutional methods, such as AGCRN, D²STGNN, on small-scale datasets have been suboptimal since smaller datasets may lack sufficient spatiotemporal patterns spatial modeling
reliably train a single adaptive graph topology. In contrast, approaches such as STNorm and STID have achieved impressive results, as their designs allow them to better capture temporal spatiotemporal patterns on limited data. Similarly. STWave with graph wavelet attention learn the underlying graph structure has also demonstrated compelling performance on certain smaller datasets. Our proposed method, however, achieves universally superior predictive accuracy, outperforming almost every baselines. This improvement can be attributed the introduction of novel mixture-of -ottuur--o--dpptiveee cereptive-raah---e graph-expe expert module. This module enables data-driven discovery of diverse underlying spatiotemporal graph topologies, thereby facilitating more precise spatiotemporal modeling.
Furthermore, also report the performance comparison of MAGE on large-scale mobile datasets Shanghai Mobile with the 3,042 nodes and Milan Internet with 10,000 nodes. As shown in Table 6, thanks to the linear-complexity yet highly expressive Mixture-of-AAaptive -aaaau------aapti----raaa wwwpdddvvv---a----eepepe Graph-Experts (MAGE) structure, our approach maintains a clear lead over all competitors on these large-scale mobility benchmarks. However, quadratic-complexin adaptive graph convolution meth as AGCRN, GWNet, D²STGNN, and Transformer based graph learning models, such as STAEformer, can not be deployed on datasets of this scale due their limited scalability. Even the complexty GNN model struggles to match the performance of the classic MLP-based approach RPMixer, owing to its inherent low-rank limitations. In stark contrast, the efficient and high-performing MAGE module within our framework is able to deftly capture diverse and meaningful spatiotemporal latent graph structures, even on these massive datasets. This breakthrough scalable spatiotemporal modeling testament to the elegance and power of our proposed approach.
26

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Table 5: Performance comparisons PeMS series datasets. The best and second best mean perfor- mance are in corresponding colors. The marker indicates baseline is not applicable to this dataset due the absence of key metadata (e.g., latitude and longitude). All experimental results the
average five independent runs.
Method
PcMIN3
PoMI04 RMSE
PeMSO7 RASE
PeMSUS RMSE
PcMls-ay RYSE
MAR
RMSE
MAPE
MAE
MAPE
MAR
MAPR
MAE
MAPΕ
MAE
MAPE
STOCK
17.04
20.62
17.36
19:27
30.85
13.16
21.89
35.64
9.45
15.72
24.93
10.64
1.74
3.76
4,06
DOCRN
15.09
26.02
16.05
18.68
30.21
13.04
20:34
33.08
8.71
14:34
23.53
9.48
1.64
3.67
3.66
AGERN
15.00
26.33
15:25
19:25
31.10
13.00
20.40
34.24
8.82
15.54
24.77
10:15
1.05
3.65
3.73
GWNNt
16.26
25.35
15. 13.38
18.81
$0.20
13.06
19.92
12.84
3.82
14:20
23.13
9.53
1.61
3.61
1.61
MITGNNN
15.31
25.05
15.04
19.20
31.81
13.26
20:97
14:30
3.90
15.22
24.09
9.89
1.63
3.66
1.60
STNom
15.82
26.48
15.00
19:44
31.24
13.42
21.23
34.54
3.96
15394
25.05
10.01
1.65
3.66
3.75
STID
15.37
26.39
16.57
18.29
29.74
12.45
10.54
32.54
3.28
14.19
23.28
9:25
1.70
3.86
3,91
XPMiser
16.19
25.91
15.96
21.31
33.56
14.88
23.95
38.77
10.63
17:33
27.47
11.31
3.91
4.36
4.27
BigST
15.30
25.77
16.54
18.42
29-96
12.92
20.31
33.57
3.57
14.19
23.26
9:24
1.03
3.58
3.77
CiSNa
15:43
25.30
15: 20
00
$0.35
13.17
20.71
33.50
3.72
15.10
23.99
9.65
1,66
3.58
3.83
STWiwe
14,89
26.80
15.
18.69
30.50
12:67
20.11
33.47
3.40
15.74
23.45
8,99
1.65
3.70
3.74
STAElense
15.27
26.76
15.85
18.78
30.30
13.06
20.09
33.36
9.41
14.17
23.38
9.18
1.65
3.61
3.75
DASTGNN
14,84
25.41
15.17
18.61
30.0
12.82
20.3 33
38.03
8.73
14.36
23.46
9:32
1.62
3.89
3.68
PrechTTT
7
7
1.62
3.65
3.67
Ours
14.72
23.33
14.07
18.16
30.16
12.64
19.49
30.00
8.25
13,66
25014
909
1.39
3.35
3.60
Table 6: Performance comparisons on large-scale mobile traffic datasets. The best and second best mean performance are in corresponding colors. The marker indicates baseline incur out-of-memory issues minimum batch size. All experimental results the average of five independent runs.
Shanghai Mobile
Milan-Internet
Method
MAE
RMSK
MAPE
MAR
RMSE
MAPE
STOCN
0.9607
1.7199
41.52
79.31
278.22
133.16
DGCRN
AGCRN
GWNet
0.9495
1.7337
39.32
46.40
159.78
58.56
MTGNN
0.9494
1.7131
40.72
66.97
230.80
117.67
STNorm
0.9735
1.7435
42.77
91.27
286.29
133.40
STID
0.9528
1.7094
40.62
47.24
152.97
50.29
RPMixer
1.0982
1.7852
53.25
44.90
140.79
55.74
BigST
0,9528
L.7079
41.38
46.44
143.73
63.60
GSNet
0.9541
1.7099
41.98
57.19
174.72
94.59
STWave
STAEformer
DºSTGNN
PatchTTG
0.9646
1.7265
40.32
54.02
180,52
59.40
Ours
0.9356
1.6832
38.44
43.08
123.90
42.59
D.3
Ablation Study
D.3.1 Ablation Study on PeMS Series Datasets and Large-scale Mobility Datasets
We design the same following variants of our model to validate the soundness of the main component of our model on PeMS Series Datasets and Large-scale Mobility Datasets: 'w/o PE' removes all the spatiotemporal position encoding embedding; 'w/o SE' uses only feedforward networks as model backbone without spatial encoder; 'w/o Multi leverages only one adaptive graph expert with K 1; 'w/o LB' reduces the load balanced optimization strategy in MAGE; "w/o Sparse' sums up all output of alternative graph convolution. The ablation study results presented in Figure 4 provide compelling insights. The performance degradation observed both the w/o LB' and w/o Sparse' variants clearly indicates that sparse and balanced graph convolution operations outperform their dense counterparts and those without balancing, respectively. Furthermore, the "w/o PE' variant also suffers from higher forecasting errors. This underscores the importance of the learnable spatiotemporal position encoding, which enables the model to extract valuable and generalizable knowledge during the training process. Interestingly, the "w/o SE' variant achieves the worst performance among all. This finding highlights the crucial role played by our mixture-of of-urr-----daptive graph convolution module in guiding the model to effectively recognize the dynamic spatiotemporal dependencies between nodes.
D.3.2 Ablation Study on Spatiotemporal Position Encoding
In this section, we conduct additional ablation study on spatiotemporal position encoding. Concretely, we construct multiple variants of MAGE in combinatorial ablation experiments by utilizing different combinations of spatiotemporal position embedding: 'w Pss" only with spatial position embedding. "w Pr is only with timestep-of-day position embedding. "w Pp is only with day-of- week position
27

---

14.7
OutS
W Ps
WPP
P
wb
PMN5-T PERE
0.9356
All Manic
wio
wio SB
18.16
LargeST-SD
LargeST-GRA
LargoS-GGLA
LargoST-CA
Spurre
PeMisR
Ours
19.53
PUMI--O
1366
XTrratlic
UrbankV
Shanghai Mobile
Milan-Intennt
(a) Ablation Study PeMS series dataets and large-scale mobility datasets.
(b) A bllaion study different spatiotemporal position encodings.
Figure 4: (a) Ablation Study on PeMS series dataets and large-scale mobility datasets. (b) Ablation
study on different spatiotemporal position encodings.
embedding, As shown in Fig 4 (b), the addition of the spatial position embedding 'w P has indeed led to significant performance gains, it enables the model to better emphasize and model the spatial positioning information. Both types of temporal position embeddings 'w Pr' and 'w PD have also proven valuable in helping the model capture temporal patterns more effectively. Notably, the 'w/o PE' variant, which lacks the spatiotemporal position encoding, exhibits the worst performance. This finding underscores the indispensable nature of the spatiotemporal position encoding in our framework. By seamlessly integrating the spatial and temporal position cues, the model is able to develop a more comprehensive understanding of the underlying data structures and dynamics.
D.4 Experimental Evaluation of the Negative Role of ReLU in Adaptive Graph Learning
In our study, we theoretically prove that ReLU operation introduces additional edge noise in adaptive graph convolution in Appendix A. In this section, we empirically evaluate the negative effectiveness of ReLU in adaptive graph convolution. Concretely, we construct variants "w/o ReLU' that reduces the ReLU operation before softmax normalization in the construction of adaptive graph of three classic STGNNs baselines: AGCRN [9], GWNet [7] and D²STGNN [23]. We conduct performance comparison experiments on deployable datasets in four domains: LargeST-SD, Electricity, Know and Beijing Weibo. We still report the average results on five experiments. As shown in Table 7, the 'w'o ReLU' variants that reduces ReLU operation in adaptive graph construction gain better performances in all the datasets due to less edge noise generating proved in the Theorem 1 in Appendix 1, possessing a relative improvement of at most 8.37%. Intriguingly, we have further
Table 7: Performance experiments on evaluating the negativity of ReLU in the adaptive graph convolution. We report the average results in five experiments. indicates the relative percentage decreasing regarding each methods itself.
Methods
LargeTT-DD
Electricity
KnowAir RMSE
Beijing Weibo
RMSE
MAR
RMSK
MAPE
MAPR
MAR
RMSE
MAPE
AGURN
33,63
13.8
211.5
1847
1695
16.34
24.81
63.26
0.8505
1.6998
53.
wlo RaLU 18.29
18,18
1333
210.0
1841
15.59
16.88
24.36
61.09
08481
1.6922
33 a315
GWNet
18.07
29.9
12.70
200.3
1820
13.48
15.49
23.85
56.73
0.8315
16777
31.74
ReLU 17,97
1010 29.33
1221
199.0
1755
13.23
23.75
56.63
0,8292
1.6665
30.88
28.60
12.15
224.8
2110
17:16
15.39
24,31
55.41
0.8189
1.7216
31.89
ReLU pugee 28
40w. g2.00
212.6
sos 2016 40% 17.
m15 24 16 a.0.
53
0.8346,
11 cars 1.7208
2...%
35
investigated the impact of incorporating ReLU activations within the adaptive graph model on the above datasets. As shown in Table the results reveal that the introduction of ReLU has detrimental effect on the training process, significantly increasing the average number of epochs required to converge. This observation aligns with our theoretical understanding that ReLU introduces undesirable edge noise into the adaptive graph generation process. This noise inherently impairs the model's s ability to capture the true spatiotemporal dynamics, thereby compromising its generalization performance. Not only does the ReLU-induced edge noise limit the upper bound of the model's
28

---

achievable accuracy, but it also adversely impacts the training convergence speed, in some cases slowing it down by more than factor of two. This is a crucial finding, as training efficiency is paramount for larger-scale real-world applications. These insights underscore the importance
Table 8: Average convergence epochs on evaluating the negativity of ReLU in the adaptive graph convolution in five experiments. The maximum allowable epochs are 300. indicates the relative percentage decreasing regarding each methods.
Datasets
AGCRN
w/o ReLU
GWNet
ReLU
D'STGNN
ReLU
LargeST-SD
216
137 00%
215
201 10.16.00%
207
186
Electricity
300
287
287.4
208
185 185 12.449% 43%
56
5247.69%
KnowAir
19.9188
34
34 3410.09% 10.0
36
33
Beijing Weibo
78
75
156
13.
47
43 30%
of principled architectural design choices when developing advanced spatiotemporal modeling frameworks. By carefully avoiding such pitfalls, our proposed MAGE approach is able to maintain its remarkable performance and training stability, even on the most challenging large-scale mobility datasets. Based on this, we have shown both theoretically and empirically that ReLU can have side effects on graph learning.
D.5 Intrinsic Preference Study of Linear and Full-rank Adaptive Graph
Building on the Pareto-front analysis in Section 5.8, we observed that a pure linear-adaptive graph already outperforms- and is markedly faster than-any mixture that includes of full-rank adaptive graphs. This raises a natural follow-up question: if the model were free to decide, which family would it actually prefer? To answer it, we replace the global load-balancing constraint with an inner-balance mechanism that enforces equal activation only within each family (linear full-rank) while letting the router autonomically allocate total capacity between the two under equal 8:8 configuration. As shown in Fig. 5, the model self-assigns 94.7% 7% of its routing mass linear experts and only 5.3% to full-rank ones, yielding both higher accuracy and better generalization. This behavior suggests that full-rank adaptive graphs are prone to overfitting and introduce redundant complexity, further validating the efficiency and representational sufficiency of the linear expert formulation adopted in MAGE.
small fraction
Linear Full
Linear Full
28.75
11.31
MAE RMSE
28.6
Inne Bolanc
28.46
5.3%
2K-4
11:
17.1
17.10
11.1
33.1%
91,7%
11.0
'Defautt
Inner Balance
MAPE (%)
(a) Performance comparison
(b) True-ratio comparison
Figure 5: The comparison between 'default' load- hal-bbalncing approach in MAGE and 'Inner balance' approach the equal ratio setting 8:8.
D.6 You Only Convolve Once
In this section, we demonstrate that the proposed MAGE achieves optimal performance with only a single graph convolution step. To this end, we construct variants of our model where each MAGE module is equipped with multi-step graph convolutions ranging from 1 to 10 layers. The results, shown in Figure 6, indicate that additional convolutional layers introduce only marginal computational overhead without yielding any significant performance gains. This observation confirms the strong expressive power of the learned multi-expert adaptive graph topology capturing complex spatiotemporal dependencies.
29

---

LagjeST-GBA
UtbaniV
17.6
20.50
Traning Spree Dlopesh
Speee Diopory
SAAB
1673
20.3
16.00
2000
Spaadbe poatt Rassus
16:55
2
1606
15.75
2
15.50
15.2
1200
Figure 6: Performance and efficiency comparison of different layers of adaptive graph convolution in MAGE.
E Discussion and Future Work
In this section, we discuss the limitations of the current work and outline potential directions for future research. First, our model currently uses only a single-layer MLP to model temporal dependencies. In future work, we plan to explore more sophisticated temporal architectures, such as Transformers, to better capture complex dynamic patterns in the data. Second, existing spatiotemporal datasets often lack essential auxiliary information-succt as community labels or ground-truth inter-regional connectivity- is necessary for evaluating properties like connectivity. As result, we are unable to thoroughly assess the effectiveness of the proposed adaptive graph in terms of structural consistency, expressiveness, and interpretability. Following Reviewer tEm8's ' suggestion, on collecting datasets enriched with broader contextual information to further evaluate the model's strengths in aspects such symmetry, normalization, and spectral properties. Finally, following Reviewer tEm8's s suggestion, we also intend to extend the proposed method to general graph learning tasks [77, 78], including node classification, link prediction, and graph classification, to further validate its broad applicability.
will therefore focus
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Reviews:
[{"review_id": "vWYYq1rZTh", "reviewer": "Reviewer_tEm8", "summary": "This paper introduces Mixture-of-Adaptive Graph Experts (MAGE), a framework for adaptive graph learning in spatiotemporal forecasting. The method uses a mixture-of-experts approach, where multiple learned graphs are combined using node-specific affinity weights. MAGE aims to reduce computational complexity and improve flexibility compared to traditional adaptive graph methods. The paper provides comprehensive theoretical analysis and experimental results on several real-world datasets to support its approach.", "questions": "For questions, please see the above raised Weakness 1 - Weakness 4. \n\nAside from W1-W4, I also have another question about the storyline of this paper. \nThis paper **has unclear motivation**. **The storyline in the Introduction is not convincing and lacks logical coherent, particularly in the introduction section.** The authors first claim that the commonly used ReLU activation function in adaptive graph learning disproportionately amplifies negative edge weights and reinforces noisy edges. To address this, they propose replacing ReLU with a kernel-based approximation scheme. However, they then argue that this approximation introduces a low-rank bottleneck in the learned adjacency matrix, which subsequently motivates the introduction of their Mixture-of-Experts (MAGE) framework. Thus, my question is: Was the motivation for introducing MAGE simply to counteract the side effects introduced by using a kernel-based approximation scheme as a replacement for the ReLU activation function?\nThis sequence of arguments is problematic: if the main issue with ReLU is noise amplification, it is unclear why the only possible alternative must be a kernel-based function that suffers from low-rank limitations. The authors do not sufficiently justify why other activation or similarity functions that avoid both noise amplification and low-rank issues could not be used instead. As a result, the motivation for MAGE appears ad hoc and unconvincing, undermining the overall contribution and logical flow of the paper.", "limitations": "Please see the above raised Weakness 1 - Weakness 4", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n(1) The proposed MAGE framework addresses the computational bottleneck of traditional adaptive graph learning by introducing a kernel-based approximation, which reduces the graph construction complexity from quadratic to linear with respect to the number of nodes. This design is particularly practical for large-scale spatiotemporal datasets and is supported by both theoretical analysis and empirical benchmarks.\n\n(2) By using a mixture-of-experts mechanism, MAGE allows for node-specific combinations of multiple learned graphs. This provides additional flexibility in capturing diverse and dynamic spatiotemporal dependencies, which could be beneficial in heterogeneous or complex forecasting scenarios.\n\n(3) The paper conducts extensive experiments across 17 real-world datasets from multiple domains, and includes ablation studies and efficiency comparisons. These experiments demonstrate that the method can achieve competitive, and sometimes state-of-the-art, results in both accuracy and efficiency when compared to a wide range of existing baselines.\n\n\n**Weaknesses:**\n\n**W1:** A major weakness of the paper is its oversimplified and outdated characterization of existing adaptive graph learning methods. The authors claim that dynamic graphs are typically constructed via pairwise embedding inner products, resulting in $O(N^2)$ computational complexity. However, this only describes a subset of early adaptive graph models such as AGCRN and GWNet. In fact, many recent adaptive graph construction approaches Implicit learning-based methods, avoiding explicit dense adjacency computation altogether. Furthremore, a large number of modern methods already employed top-K selection, sparse routing, or on-the-fly edge generation, achieved much lower complexity in practice [1] [2]. Therefore, presenting the $O(N^2)$ embedding inner product approach as representative of the entire field is inaccurate and ignores the significant methodological diversity and existing research progress that have been made in adaptive graph learning.\n\n\n\n**References:**\n\n\n[1] Huang, Siyuan, et al. \"Tailoring self-attention for graph via rooted subtrees.\" *Advances in Neural Information Processing Systems* 36 (2023): 73559-73581.\n\n[2] Qian, Chendi, et al. \"Probabilistic Graph Rewiring via Virtual Nodes.\" *The Thirty-eighth Annual Conference on Neural Information Processing Systems*.\n\n\n\n\n\n**W2: Limited Global Structural Consistency, Expressiveness, and Interpretability.** The design of MAGE enables each node to construct its own adaptive adjacency structure by selectively combining expert-generated graphs using node-specific affinity weights. While this approach allows for high local flexibility, it fundamentally breaks the consistency of a unified global graph structure. Unlike traditional GNNs, where all nodes share the same adjacency matrix. As a result, important global properties such as overall connectivity, cycles, community structure, or long-range dependencies may not be explicitly modeled or preserved. This lack of structural coherence can be particularly problematic in tasks where global graph topology or prior knowledge plays a crucial role.\n Additionally, the \"personalized\" and fragmented nature of the learned adjacency makes it extremely difficult to interpret or visualize the resulting graph, limiting its practical utility for scientific analysis, knowledge discovery, or downstream applications that require graph explanations. The method also lacks theoretical justification regarding its ability to model or recover key global attributes (such as symmetry, normalization, or spectral properties) that are well-studied in the GNN literature. Without systematic empirical or theoretical analysis on these aspects, it is unclear whether the proposed approach can generalize or perform robustly on problems where global graph characteristics are critical.\n\n\n\n**W3: Lack of Direct Graph-Level Validation and Analysis.** The experimental validation in this paper is largely indirect. The majority of the experimental section focuses on demonstrating that using MAGE leads to improved spatiotemporal forecasting performance on certain datasets, as measured by standard prediction error metrics. However, such improvements only provide indirect evidence of the effectiveness of the proposed graph learning method. The paper lacks more direct graph-level validation from the perspective of graph structure analysis. For example, there is no comparison of the sparsity or connectivity patterns of the graphs constructed by MAGE versus other methods, to show whether MAGE successfully preserves the most important edges. Additionally, the paper does not include any visualization or alignment of the learned graphs with the true underlying system structure at time $t$, to demonstrate that the edges constructed by MAGE meaningfully correspond to real, strong connections in the data. This omission makes it difficult to assess whether the proposed method actually learns more accurate or interpretable graph structures.\n\n\n\n**W4:** **Insufficient Empirical Evidence for the Claimed Drawbacks of ReLU Activation.** The authors claim that the use of ReLU activation in existing adaptive graph learning methods amplifies edge-level noise, resulting in the creation of spurious or unreliable connections in the learned adjacency matrix. However, this assertion is only supported by a simple theoretical analysis, without any dedicated empirical validation. The paper does not provide ablation studies, visualizations, or quantitative experiments to directly demonstrate the negative impact of ReLU-induced noise or to substantiate the practical advantages of their proposed alternative. As a result, the significance of this issue remains unconvincing.", "was_revised": true, "final_justification": "The paper introduces MAGE for efficient adaptive graph learning in spatiotemporal forecasting, with strong empirical results. The rebuttal provided additional baselines, partial graph-level evaluation, and clarification that MAGE and the ReLU-related modification are independent contributions. However, the core concerns remain: the initial framing of prior work still overlooks recent efficient adaptive graph methods; global structural consistency and interpretability are not convincingly addressed; graph-level analysis is limited and largely indirect; and the motivation linking design choices remains unconvincing. Despite the added clarifications, these fundamental weaknesses persist, so my original reject rating is upheld.", "rebuttal": "We sincerely thank you for your professional advice and the time you have dedicated. We have gained valuable insights, and based on your feedback, we can significantly enhance the quality of our paper.\n\n---\n---\n\n## **W1.**\n\nThe methods in [1, 2] are computationally efficient variants of the Transformer model, characterized by high model complexity and a large number of parameters. In contrast, the dot-product adaptive graph module requires only two learnable embeddings for pairwise dot-product calculations, making it significantly more lightweight and simplified. **This approach also offers good ease of integration and has consequently been widely adopted. For these reasons, we have chosen the dot-product adaptive graph learning method as the foundation of our research.**\n\nTo demonstrate this through experiments, we addressed a critical limitation: the works [1, 2] focus on graph representation tasks and lack the temporal modeling elements needed for spatiotemporal prediction. To ensure a fair comparison, we integrated the graph learning methods from MAGE and [1, 2] into mature spatiotemporal prediction models GWNet and AGCRN by replacing their adaptive graph modules. We then assessed their performance and complexity, under optimal performance settings, on the SD dataset. The results are summarized below.\n\n|| MAE|RMSE|MAPE|Memory (MB)|Training Speed (s/epoch)|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|GWNet| 18.07| 29.97| 12.70|8,990|83.56|\n|+STA| 19.13| 31.39| 12.83|9,737|81.18|\n|+IPR-MPNNs| 26.22| 41.49| 17.33|9,302|78.77|\n|+MAGE|**17.87**|**29.43**|**12.26**|**6,994**|**67.52**|\n\n|| MAE|RMSE|MAPE|Memory (MB)|Training Speed (s/epoch)|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|AGCRN| 18.39| 33.63|13.78|8,100|80.58|\n|+STA| 18.05| 33.08 |13.61|8,746|76.32|\n|+IPR-MPNNs| 20.45|34.51|14.41|8,443|65.94|\n|+MAGE|**18.00**|**32.21**|**13.34**|**6,810**|**59.12**|\n\nWe discovered that integrating these Transformer variants into the model actually led to decreased performance, along with a significant increase in the number of parameters and memory usage. \n\nBased on your suggestions, we will integrate these discussions and related recent advancements into the manuscript in future revisions. Thank you again for your advice.\n\n---\n---\n\n## **W2.** \n\nThank you for your valuable insights into graph learning, which will inspire our future research.\n\n> Allow me to clarify first: our research is focused solely on the specific downstream task of spatiotemporal prediction, and **does not aim to develop a general-purpose GNN applicable to various downstream tasks.**\n\nIn the field of spatiotemporal forecasting, **(1) Datasets often lack essential ancillary information necessary for evaluating properties such as connectivity, which can limit the comprehensive analysis.** **(2) The effectiveness of traditional GNNs that rely on a shared global graph structure is being questioned. Currently, adaptive graph learning methods dominate the field.** The main reasons are as follows:\n\n- **Poor-quality or even non-constructable graph topology**. Unlike general graph learning settings, graphs in most spatiotemporal prediction datasets are not naturally occurring but are constructed using secondary measures such as geographical proximity. **These distance-based graphs fail to capture other critical global properties**, such as connectivity, cyclicity, and community structure. In some cases, even global graphs derived from geographic information are unavailable due to privacy concerns.\n\n- **There is a lack of relevant prior data to validate the advantages of GNNs with shared global graphs**. Most datasets lack the necessary prior knowledge, such as community labels or actual inter-regional connectivity, which hinders the evaluation of the advantages of GNNs with unified global graph in terms of properties like symmetry, normalization, or spectral attributes. **These limitations further restrict the direct assessment of model interpretability.**\n\n> We can take a different perspective: **perhaps we can integrate the adaptive graph learning method MAGE into GNNs based on unified global graph**. \n\nIn this way, it can retain the advantages of MAGE while also inheriting the strengths of traditional GNNs. To explore this possibility, we have incorporated MAGE into GWNet and STGCN by performing multi-graph convolution using both the adaptive adjacency matrix and their global graphs. The following results demonstrate that MAGE can improve the performance of GNNs based on unified global graph, which inspires our future work in designing hybrid graph convolutional networks.\n\n|SD dataset|MAE|RMSE|MAPE|\n|:-:|:-:|:-:|:-:|\n|STGCN|19.27|33.57|13.49|\n|+MAGE|19.19|32.89|12.70|\n|GWNet|18.07|29.97|12.70|\n|+MAGE|**17.87**|**29.43**|**12.26**|\n\n---\n---\n\n## **W3.** \n\nThe ultimate goal of spatiotemporal forecasting is to improve prediction accuracy. **Beyond the core adaptive graph learning component, our MAGE model consists solely of simple fully connected layers; therefore, its performance gains are primarily attributable to the proposed adaptive graph learning method**. This is extensively validated by the ablation studies in Figure 2 of the paper. These two evidences directly prove the effectiveness of our graph learning approach. \n\nThank you for your suggestion to include additional graph evaluations. We appreciate your feedback and would like to provide the following response.\n\n- Experiment \n\nA well-structured graph topology should effectively reflect the similarity among node labels. To evaluate the quality of the learned graph topologies, we first integrate three advanced methods—MAGE, [1, 2], and the original adaptive graph learning—into GWNet, thereby isolating the evaluation from potential difference factors introduced by other model components. We then report the mean squared error (MSE) between the generated adjacency matrices and the similarity matrix derived from node labels, providing a quantitative measure of how well the learned structures capture label coherence.\n\n| SD dataset  | Ours| STA | IPR-MPNNs |Original\n|:-:|:-:|:-:|:-:|:-:|\n| Similarity MSE | **0.00135**| 0.198| 0.389|0.00838|\n\nThe adjacency matrix learned by our model shows the smallest difference with respect to the node label affinity, which provides a solid foundation for accurate prediction by the model.\n\n-  Visualization\n\nDue to the NIPS rebuttal policy prohibiting the inclusion of any images, we promise to add more visualizations of generated graphs in the final version. These cases will demonstrate the association between the learned adjacency matrices and the patterns of data distributions.\n\n---\n---\n\n## **W4.** \n\n> **We have done this experiment!** As emphasized in line 115 of the manuscript, we provide experimental evidence for this theory, with the results shown in Appendix D.6.\n\nSpecifically, we demonstrate across four datasets and multiple models employing the adaptive graph learning module that removing the ReLU activation function improves prediction performance, as shown in **Table 6**. Furthermore, as illustrated in Table 7, removing ReLU also accelerates model convergence. For convenience, we reproduce a subset of results on the SD dataset below. These findings provide strong empirical evidence supporting the effectiveness of our design choice.\n\n\n|  |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:|\n| AGCRN    | 18.39 | 33.63 |  13.78| \n| w/o ReLU   |**18.29** | **33.18**| **13.32**|\n\n|             |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:| \n| GWNet    | 18.07  | 29.97 |  12.70  |\n| w/o ReLU   | **17.97**| **29.33**|**12.21**|\n\n|             |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:|\n|D2STGNN|17.13|28.60|12.15|\n| w/o ReLU   |**16.99**|**28.46**|**12.03**| \n\n（2）We added the ReLU function to the proposed MAGE model and compared its performance with the original MAGE, as shown in the table below. This result also serves as valid experimental evidence.\n\n|  | MAE |  RMSE | MAPE |\n|:-:|:-:|:-:|:-:|\n|Ours|**16.29**|**28.04**|**10.87**|\n|+ ReLU   |16.81|28.90|11.05| \n\n---\n---\n\n## **Q1. MAGE has nothing to do with ReLU.**\n\n> MAGE is **Not** simply to counteract the side effects introduced by using a kernel-based approximation scheme as a replacement for the ReLU activation function!**\n\n**As emphasized in the abstract, introduction, and methods sections, MAGE and the ReLU noise theory improve two distinct components including in adaptive graph learning framework**—namely, RuLE and the vector dot product—and represent independent improvements. **Therefore, MAGE is not designed to mitigate any side effects that may arise from replacing the ReLU activation function with a kernel-based approximation scheme.** Specifically,\n\n- RuLE Noise Theory: This theory reveals that RuLE introduces noise, leading to performance degradation, and thus we remove it. **As demonstrated in W4, this modification does not introduce any side effects**; on the contrary, it brings performance improvements.\n\n- MAGE: MAGE is designed to address the challenge of high computational complexity in dot product operations. First, we propose kernel-based functions as efficient alternatives to dot products. Then, we introduce a sparse yet balanced mixture-of-experts system to enhance expressive power.\n\nTo further avoid confusion, we will revise the last paragraph of the introduction and the methods section to highlight the relationship between the two, namely that we have made independent improvements to the two components of dot-product-based adaptive graph learning.\n\nRef:\n\n[1] Huang, Siyuan, et al. \"Tailoring self-attention for graph via rooted subtrees.\" Advances in Neural Information Processing Systems 36 (2023): 73559-73581.\n\n[2] Qian, Chendi, et al. \"Probabilistic Graph Rewiring via Virtual Nodes.\" The Thirty-eighth Annual Conference on Neural Information Processing Systems."}, {"review_id": "d1g7vCZhqK", "reviewer": "Reviewer_r3iW", "summary": "The authors introduce an efficient sparse yet balanced mixture-of-experts strategy to tackle the complexity associated with current adaptive graph learning methods. They underpin the validity of their approach with comprehensive theoretical analyses. Compared with 14 spatiotemporal models across 17 datasets, their model demonstrates remarkable performance, while achieving significant improvements in memory efficiency, with a 10× reduction, and training efficiency, with a 20× increase.", "questions": "What is the model's potential for scalability on extremely large-scale spatiotemporal graphs?", "limitations": "Yes, the authors have discussed limitations.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\nS1. The proposed model is reasonably motivated and logically well-designed, effectively addressing various challenges step-by-step.\n \nS2. The idea of sparsing and balancing among the mixture of experts is admirable.\n \nS3. This paper presents new theoretical insights.\n\nS4. Extensive experiments including 17 datasets and 14 baselines prove the competitive performance and efficiency of their model. \n\n\nWeakness:\n\nW1. The authors have not provided an analysis or explanation for why PatchSTG fails to run on certain datasets. \n\nW2. Spatiotemporal forecasting is a subset of time series, necessitating further discussion around additional time series forecasting models, such as PatchTST and TimeMixer. \n\nW3. Is it possible to achieve performance gains by expanding the number of experts?\n\nRef:\n\n[1] Nie, Y., Nguyen, N. H., Sinthong, P., & Kalagnanam, J. (2022). A time series is worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2211.14730.\n\n[2] Wang, S., Wu, H., Shi, X., Hu, T., Luo, H., Ma, L., ... & Zhou, J. (2024). Timemixer: Decomposable multiscale mixing for time series forecasting. arXiv preprint arXiv:2405.14616.", "was_revised": true, "final_justification": "The authors' response has addressed my concerns.", "rebuttal": "> Thank you very much for your kind recognition. Your time and effort have been invaluable in significantly improving the quality of our paper. We truly appreciate your support and constructive feedback.\n\n## **W1. PatchSTG issues**\n\nWe apologize for any confusion caused. PatchSTG model requires the latitude and longitude information of each node as additional input to determine which patch each node belongs to. However, due to the potential absence of such geographical data in publicly available datasets, the input requirements of the PatchSTG model cannot be met. Consequently, it fails to execute on certain datasets.\n\n---\n---\n\n## **W2. Time series baselines**\n\nTime series models perform poorly in spatiotemporal prediction tasks because they only model dependencies between time steps while ignoring the crucial spatial dependencies among nodes [1].\n\nTo address your question, we compared the performance of the MAGE model with time series models based on different architectures, including the Transformer-based PatchTST, the convolution-based TimeMixer, and the MLP-based DLinear, on the SD and KnowAir datasets, as shown below.\n\n| Model       |        |      SD |         |             |      KnowAir |              |\n|-------------|--------|---------|---------|-------------|--------------|--------------|\n|             |    MAE |    RMSE |    MAPE |         MAE |         RMSE |         MAPE |\n| PatchTST    | 48.50  | 74.20   | 30.84   | 19.04       | 29.28        | 61.66        |\n| TimeMixer   | 40.68  | 59.89   | 35.22   | 16.78       | 25.85        | 65.38        |\n| DLinear     | 49.82  | 75.43   | 39.96   | 17.06       | 26.20        | 64.53        |\n| MAGE(Ours)  |**16.29**  | **28.04**  | **10.87**  | **15.36**      | **23.42**        |**52.77**       |\n\nThe results demonstrate that MAGE still exhibits the best performance on the aforementioned datasets, validating the effectiveness of our model.\n\n[1] Shao Z, Wang F, Xu Y, et al. Exploring progress in multivariate time series forecasting: Comprehensive benchmarking and heterogeneity analysis[J]. IEEE Transactions on Knowledge and Data Engineering, 2024.\n\n---\n---\n\n## **W3. Performance of expanding more experts.**\n\nBlindly increasing the total number of experts does not improve performance; on the contrary, it may lead to overfitting the training data and result in decreased generalization ability. According to our matrix theory, once the number of experts exceeds a certain threshold, the model’s expressive power reaches its upper limit. As shown in Figure 2(b) in Section 5.3 of this paper, we systematically varied both quantities and reported the corresponding results. The experimental findings indicate that when the number of experts is too large, model performance actually drops.\n\n---\n---\n\n## **Q. Potential for scalability**\n\nMAGE exhibits a computational complexity that scales linearly with the number of nodes, demonstrating strong theoretical scalability. Furthermore, we evaluate MAGE on XTraffic—the largest publicly available spatiotemporal dataset to date, containing 16,972 nodes. The results show that MAGE still achieves the best accuracy, further validating its excellent scalability. In contrast, several more complex models fail to run due to their high computational demands. For convenience, we present below the performance comparison with several state-of-the-art models.\n\n|    |    MAE |    RMSE |    MAPE |\n|-------------|--------|---------|---------|\n| PatchSTG    |10.63|20.86| 19.41 | \n|STID|  11.62 | 22.41 | 19.84 | \n| RPMixer| 16.68| 43.64| 32.74|\n| MAGE(Ours)  | **10.24**  |**20.48** |**17.92**|"}, {"review_id": "zQWpZ7imwx", "reviewer": "Reviewer_hrJD", "summary": "This paper tackles the issues of edge noise and high complexity in traditional adaptive graph convolution methods by introducing a sparse mixture-of-experts linear graph learning approach. Theoretical analysis is provided to demonstrate the method’s effectiveness. Extensive experiments and comparisons with baseline models validate the efficacy and efficiency of the proposed model.", "questions": "Q1: Is there a way to avoid multiple experts learning redundant knowledge?", "limitations": "yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\nS1: This paper presents an novel integration of the mixture-of-experts mechanism with spatiotemporal graph learning, and designs some novel techniques with solid theoretical contributions.\n\nS2: The experiments conducted on 15 datasets are informative and comprehensive. And a 20-fold increase in efficiency is impressive.\n\nS3: The authors have open-sourced the code to facilitate reproducibility. Focusing on efficient model for large-scale data is a cutting-edge research direction.\n\n**Weakness:**\n\nW1: The numerical formatting in Table 2 is inconsistent. Some values use separators while others do not. There are also spelling errors. For example, in line 289, there is an extra space inside the parentheses: (60%). Figure 2(a) is suggested to be presented in tabular form to improve readability.\n\nW2:  The paper argues that the RULE function may introduce noise, but further experimental validation is needed to support this claim. \n\nW3: Some important spatiotemporal forecasting benchmarks should be included for further comparison, such as LLGformer [1].\n\nW4: The existing mixture-of-experts models in the spatiotemporal data domain should be discussed [2][3]. \n\n[1] Jin, D., Huo, C., Shi, J., He, D., Wei, J., & Yu, P. S. (2025, April). Llgformer: Learnable long-range graph transformer for traffic flow prediction. In Proceedings of the ACM on Web Conference 2025 (pp. 2860-2871)\n\n[2] Jiang W, Han J, Liu H, et al. Interpretable cascading mixture-of-experts for urban traffic congestion prediction. Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2024: 5206–5217.\n\n[3] Li S, Cui Y, Zhao Y, et al. ST-MoE: Spatio-temporal mixture-of-experts for debiasing in traffic prediction. Proceedings of the 32nd ACM International Conference on Information and Knowledge Management. 2023: 1208–1217.", "was_revised": true, "final_justification": "The authors have adequately addressed my concerns, and I choose to maintain a positive evaluation.", "rebuttal": "> Thank you very much for your kind recognition. Your time and effort have been invaluable in significantly enhancing the quality of our paper. We truly appreciate your support and constructive feedback.\n\n## **W1. Typos of numerical formatting**\n\nThank you for this valuable suggestion. In the camera-ready version we will perform a meticulous proof-read of every numeric entry in the tables to ensure a uniform, consistent format throughout.\n\n---\n---\n\n## **W2. Experiments of edge noise induced by ReLU**\n\nWe conducted a comprehensive experimental study (reported in **Appendix D6: Experimental Evaluation of the Detrimental Effects of ReLU in Adaptive Graph Learning**), which confirms that removing the ReLU activation consistently improves both prediction accuracy and convergence speed. Specifically, we selected four representative models—GWNet, AGCRN, and D$^2$STGNN—that incorporate the adaptive graph convolution module, and evaluated them on four real-world datasets spanning diverse domains. The results provide strong empirical support for our theoretical claims. For your convenience, we reproduce a subset of these results below.\n\n|  |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:|\n| AGCRN    | 18.39 | 33.63 |  13.78| \n| w/o ReLU   |**18.29** | **33.18**| **13.32**|\n\n|             |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:| \n| GWNet    | 18.07  | 29.97 |  12.70  |\n| w/o ReLU   | **17.97**| **29.33**|**12.21**|\n\n|             |    MAE |    RMSE |    MAPE |\n|:-:|:-:|:-:|:-:|\n|D$^2$STGNN|17.13|28.60|12.15|\n| w/o ReLU   |**16.99**|**28.46**|**12.03**| \n\n---\n---\n\n## **W3. More Spatiotemporal Baselines**\n\nIn the main experiments, we have already covered **17** open-source and publicly available advanced spatiotemporal models. Unfortunately, the code for LLGformer [1] has not been released. To address your concern, we introduce STGformer [2] as an additional baseline and evaluate it on four datasets across three domains. The results are reported below.\n\n| Model      |        | SD      |         |        | CA      |         |             | KnowAir      |              |             | UrbanEV      |              |\n|:------------:|:--------:|:---------:|:---------:|:--------:|:---------:|:---------:|:-:|:--:|:--:|:--:|:---:|:--:|\n|            |    MAE |    RMSE |    MAPE |    MAE |    RMSE |    MAPE |         MAE |         RMSE |         MAPE |         MAE |         RMSE |         MAPE |\n| STGformer  | 18.90  | 31.65   | 12.76   | 21.90  | 35.29   | 17.01   | 15.91       | 24.43        | 58.11        | 5.13        | 11.09        | 18.51        |\n| MAGE       | **16.29**  | **28.04**   | **10.87**   | **17.37**  | **29.37**   | **12.47**   | **15.36**       | **23.42**        | **52.77**        | **4.95**        | **11.00**        | **17.43**        |\n\n\n\n---\n---\n\n## **W4. More discussion about MoE in Spatiotemporal Learnings**\n\n- **CP-MoE** [3] targets the *spatiotemporal congestion classification* task. Its MoE design relies on two graph experts that separately learn periodic and non-periodic congestion patterns. \n\n- **ST-MoE** [4] tackles *learning imbalance* in spatiotemporal tasks by stacking multiple STGNNs into an MoE network, allowing each node to select an appropriate STGNN.\n\nIn contrast, our MoE module aims to \"**restore the rank upper bound of linear adaptive graph convolution**\": we employ a minimal set of graph convolutions to mitigate the inherent low-rank bottleneck commonly found in linear-complexity designs. Additionally, **we introduce a sparse and balanced modulation strategy to regulate the learning behavior of individual experts**. As a result, **both the objective and mechanism of our approach are fundamentally distinct from those of the aforementioned works**. Unfortunately, no open-source implementations of CP-MoE or ST-MoE are currently available, making direct performance comparisons infeasible. We commit to including a detailed discussion of these models in the next version of the manuscript.\n\n\n\n\n---\n---\n\n## **Q1. Avoiding Knowledge Redundant in Experts.**\n\nWe achieve this goal by integrating a load balancing mechanism to ensure that each graph expert is utilized evenly during training, while sparse gating enables each node to select only the most suitable experts. Together, these mechanisms encourage each expert to focus on capturing distinct underlying patterns. Additionally, based on matrix theory, we carefully control the number of experts to further prevent knowledge redundancy.\n\nRef: \n\n[1] Jin, D., Huo, C., Shi, J., He, D., Wei, J., & Yu, P. S. (2025, April). Llgformer: Learnable long-range graph transformer for traffic flow prediction. In Proceedings of the ACM on Web Conference 2025 (pp. 2860-2871)\n\n[2] Wang H, Chen J, Pan T, et al. Stgformer: Efficient spatiotemporal graph transformer for traffic forecasting[J]. arXiv preprint arXiv:2410.00385, 2024.\n\n[3] Jiang W, Han J, Liu H, et al. Interpretable cascading mixture-of-experts for urban traffic congestion prediction. Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2024: 5206–5217.\n\n[4] Li S, Cui Y, Zhao Y, et al. ST-MoE: Spatio-temporal mixture-of-experts for debiasing in traffic prediction. Proceedings of the 32nd ACM International Conference on Information and Knowledge Management. 2023: 1208–1217."}, {"review_id": "NqqQfSdquj", "reviewer": "Reviewer_41Uo", "summary": "This paper studies the problem of GNN-based spatiotemporal forecasting. Motivated by the deficiency of RELU in modeling dynamical graph topology used behind GNNs, this paper removes RELU and proposes a sparse yet balanced MOE strategy for efficiently approximating the full-rank Linear Adaptive Graphs. Extensive experiments on a large number of benchmark datasets show the superior performance of the proposed method against a sizeable collection of baselines.", "questions": "Please see the weaknesses.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- This paper is well motivated by illustrating the deficiency of RELU in modeling dynamical graph topology used behind GNNs.\n- The proposed method is technically sound.\n- The experiments compare the proposed method against a large collection of baselines on a number of benchmark datasets.\n\nWeaknesses:\n- Some design choices are not well justified, particularly the necessity of the kernel-inspired approximation. What if you directly apply MOE on the full-rank linear adaptive graphs, which completely avoid the low-rank bottleneck? Theoretically, this will be more expressive than your proposed method at the cost of higher computational cost. This trade-off should be rigorously discussed and studied in the experiments. What is the Pareto frontier of this trade-off?\n- The proposed MAGE can be regarded as a base building block, which could be integrated with other spatiotemporal GNN models. Its advantages should be better justified by integrating with several other SOTA models. It would be interesting to see whether the performance improvement persists in these cases.", "was_revised": true, "final_justification": "After the rebuttal and discussion, I kept my positive score.", "rebuttal": "Thank you very much for your kind recognition. Your time and effort have been invaluable in significantly improving the quality of our paper. \n\n---\n\n## **W1.**\n\n> a. The necessity of the kernel-inspired approximation\n\nOur motivation is to design a linear yet expressive adaptive graph structure, and these two goals are achieved through the two components of MAGE (MOE+kernel linear approximation). Specifically, **linearity is achieved via a kernel-inspired approximation method**, while the subsequent customized sparse and balanced MOE strategy helps restore the model’s rank to its upper bound (full rank), thus enhancing its representational power.\n\n\n> b. Only MAGE simultaneously satisfies both linear and full-rank properties in existing adaptive graphs.\n\nIn fact, existing adaptive graph either possess full-rank characteristics (e.g., naive dot-product-based adaptive graph methods) but suffer from quadratic complexity, or emphasize linear complexity (e.g., relying solely on kernel-inspired dot-product approximations) yet are limited by low-rank constraints. Thus, **there are another full-rank linear adaptive graph case.** However, your question leads to another interesting one: what would happen if MOE were applied to a ~~linear~~ full-rank adaptive graph?\n\n> c. What if you directly apply MOE on the ~~linear~~ full-rank adaptive graphs, which completely avoid the low-rank bottleneck?\n\nTo address this problem, we designed a MAGE variant called “MOE+ Nai”, where we combine a sparse and balanced MoE strategy with a naive (full-rank) adaptive graph. The hyperparameter for the number of experts is kept the same. We report its performance and efficiency on the SD dataset.\n\n\n| SD | MAE  | RMSE | MAPE | Memory  | Training Speed |\n|---------------|------|-------|------|----------|---------------|\n| MAGE (MOE+Linear kernel approximation)| **16.29** | **28.04** | **10.87** | **3,662 MB**| **22 s/epoch** |\n| MOE+Nai | 16.52 | 28.35 | 10.91 |4,308 MB| 46 s/epoch|\n\nDue to the quadratic complexity of the naive adaptive graph, this variant failed to improve performance and instead increased memory usage and training time. This is because our sparse and balanced MoE strategy is specifically designed for linear kernel adaptive graphs: by combining expert models that can capture different graph patterns, it enhances representational capacity and effectively achieves a full-rank approximation. However, when MoE is applied to a full-rank graph (which has already reached the upper bound of rank), it introduces redundant information, thus degrading learning performance. We provide further explanation in Section d(2).\n\n\n\n> d. This trade-off should be rigorously discussed and studied in the experiments. What is the Pareto frontier of this trade-off?\n\n(1). To address your question, we designed a set of experiments with 16 fixed graph experts, where each expert corresponds to an adaptive graph. According to the paper, 16 experts can achieve optimal performance. We then varied the mixing ratio of linear kernel adaptive graphs and naive adaptive graphs (full-rank), maintaining the original balanced and sparse expert activation constraints. The experimental results are as follows:\n\n| Linear : Full | MAE  | RMSE | MAPE | Memory  | Training Speed |\n|---------------|------|-------|------|----------|---------------|\n| 16:0 (Ours)          | **16.29** | **28.04** | **10.87** | **3,662 MB**   | **22 s/epoch**          |\n| 12:4          | 16.29 | 28.22 | 10.88 | 3,696 MB   | 24 s/epoch          |\n| 8:8           | 17.10 | 28.75 | 11.31 | 3,860 MB   | 30 s/epoch          |\n| 4:12          | 16.53 | 28.29 | 11.01 | 3,998 MB   | 35 s/epoch          |\n| 0:16 (MOE+Nai)         | 16.52 | 28.35 | 10.91 | 4,308 MB   | 46 s/epoch          |\n\nWe observed that gradually increasing the proportion of full-rank adaptive graphs does not provide additional performance benefits aside from reducing computational efficiency. Therefore, we believe that the original MAGE module design has achieved a Pareto optimal state.\n\n\n(2). During this process, we observed an interesting phenomenon regarding the model’s expert selection behavior—its preference for linear adaptive graph experts over naive gadaptive raph experts. We used an 8:8 balanced ratio as a study example. Initially, the balance strategy forced the model to equally activate all 16 experts. In the variant, we introduced the concept of 'Inner Balance', allowing the model to adaptively choose between the linear graph expert family and the full-rank graph expert family, while maintaining balance in the activation of individuals within each family. \n\n|   | MAE  | RMSE  | MAPE  | Turth Ratio (Linear kernel vs Full-rank)    |\n|:---:|:--:|:-:|:-:|:-:|\n| Default | 17.10 | 28.75 | 11.31 | 53.1% : 46.9%   |\n| Inner Balance     | **16.66** | **28.46**  | **11.00** | 94.7% : 5.3%   |\n\nThe results indicate that the Inner Balance variant not only improved performance but also revealed the model’s strong preference for linear kernel graph expert (94.7% vs. 5.3%). The default global balance (53.1% vs. 46.9%) inadvertently forces the model to retain overfitting full-rank experts, which can harm generalization ability. This further provides evidence of the incompatibility between MOE and full-rank adaptive graphs.\n\n---\n\n## **W2.**\n\n\nThat’s a very innovative question. Using the SD dataset as an example, we integrated MAGE into the following four advanced spatiotemporal learning models: AGCRN, GWNet, STGCN, and STID. We reported their average performance over 12 time steps. \n\n\n| Model| MAE| RMSE| MAPE|\n|-------------|--------|--------|--------|\n| STGCN       | 19.27  | 33.57  | 13.49  |\n| +MAGE       | **19.19**  | **32.89**  | **12.70**  |\n\n| Model| MAE| RMSE| MAPE|\n|-------------|--------|--------|--------|\n| GWNet       | 18.07  | 29.97  | 12.70  |\n| +MAGE       | **18.13**  | **29.43**  | **12.26**  |\n\n| Model| MAE| RMSE| MAPE|\n|-------------|--------|--------|--------|\n| STID        | 18.03  |30.85   |12.18|\n| +MAGE       | **17.83**  |**30.42**    |**12.07**|\n\n| Model| MAE| RMSE| MAPE|\n|-------------|--------|--------|--------|\n| AGCRN       | 18.39  | 33.63  | 13.78  |\n| +MAGE       | **18.00**  | **32.21**  | **13.34**  |\n\nWe observe that MAGE is able to improve the performance of these models to varying degrees, which demonstrates the good applicability of our method."}]

Author rebuttal:

Meta-review: This paper proposes MAGE for efficient spatiotemporal forecasting. The paper explores the validity of their approach with comprehensive theoretical analyses and compares with multiple spatiotemporal models across different datasets, and displays remarkable performance. The theoretical analysis of this paper is rigorous and the experimental evaluations are sufficient to justify the statements.

I'm leaning toward accepting the paper.
Decision: Accept (poster)