CSG-ODE: ControlSynth Graph ODE For Modeling Complex Evolution of Dynamic Graphs
Authors: Zhiqiang Wang
Xiaoyi Wang
Jianqing Liang
Venue: ICML 2025 poster (2025)
Keywords: Graph Neural Network
Dynamic Graph
Graph Neural ODE
VAE
Primary area: deep_learning->graph_neural_networks
TL;DR: This work introduces ControlSynth Graph ODE (CSG-ODE), a novel approach enhancing dynamic graph modeling via inter-node importance weighting and nonlinear evolution, with its stable extension (SCSG-ODE) achieving superior performance and stability.

Abstract: Graph Neural Ordinary Differential Equations (GODE) integrate the Variational Autoencoder (VAE) framework with differential equations, effectively modeling latent space uncertainty and continuous dynamics, excelling in graph data evolution and incompleteness. However, existing GODE face challenges in capturing time-varying relationships and nonlinear node state evolution, which limits their ability to model complex dynamic graphs. To address these issues, we propose the ControlSynth Graph ODE (CSG-ODE). In the VAE encoding phase, CSG-ODE introduces an information transmission-based inter-node importance weighting mechanism, integrating it with latent correlations to guide adaptive graph convolutional recurrent networks for temporal node embedding. During decoding, CSG-ODE employs ODE to model node dynamics, capturing nonlinear evolution through sub-networks with nonlinear activations. For scenarios or prediction tasks that require stability, we extend CSG-ODE to stable CSG-ODE (SCSG-ODE) by constraining weight matrices to learnable anti-symmetric forms, theoretically ensuring enhanced stability. Experiments on traffic, motion capture, and simulated physical systems datasets demonstrate that CSG-ODE outperforms state-of-the-art GODE, while SCSG-ODE achieves both superior performance and optimal stability.

ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
CSG-ODE:
Zhiqiang Wang'
Xiaoyi Wang'
Jianqing Liang'
Abstract
1. Introduction
With the rapid development of fields such as social net- works, traffic flow prediction, and financial market analysis, dynamic graph representation learning has gradually be- come one of the important research directions in the field of graph learning (Yang et al., 2024). A dynamic graph refers to a graph structure in which the states of nodes and edges, as well as the graph topology, evolve over time. In these systems, the interactions and relationships between nodes are time-varying, presenting both significant challenges and opportunities for model development.
Graph Neural Ordinary Differential Equations (GODE) integrate the Variational Autoencoder (VAB) framework with differential equations, ef- fectively modeling latent space uncertainty and continuous dynamics, excelling in graph data evo- lution and incompleteness. However, existing GODE face challenges in capturing time-varying relationships and nonlinear node state evolution, which limits their ability to model complex dy- namic graphs. To address these issues, we pro- pose the ControlSynth Graph ODE (CSG-ODE). In the VAE encoding phase, CSG-ODE introduces an information transmission-base inter-node im- portance weighting mechanism, integrating it with latent correlations to guide adaptive graph con- volutional recurrent networks for temporal node embedding. During decoding, CSG-ODE em- plays ODE to model node dynamics, capturing nonlinear evolution through sub-networks with nonlinear activations. For scenarios prediction tasks that require stability. we extend CSG-ODE to stable CSG-ODE (SCSG-ODE) by constrain- ing weight matrices to learnable anti-symmetric forms, theoretically ensuring enhanced stability. Experiments on traffic, motion capture, and simu- lated physical systems datasets demonstrate that CSG-ODE outperforms state-of-the-art GODE, while SCSG-ODE achieves both superior perfor- mance and optimal stability.
In recent years, with the increasing popularity of Graph Neural Networks (GNN) (Scarselli et al., 2008), more and more dynamic graph representation learning methods have begun to leverage GNN to handle the temporal information in dynamic graphs By introducing strategies such as tem- poral window mechanisms (Zhu et al., 2023) and temporal encoding (Chen et al., 2024), GNN can effectively enhance the model's 's ability to process dynamic graph data and cap- ture the temporal dynamics of nodes and edges. However, many existing methods still assume that the complete state of all nodes available each time point. This assumption is often difficult to hold in
applications.
LatentODE (Rubanova et al., 2019) effectively addresses the challenge of modeling irregularly sampled time series by introducing latent ordinary differential equations (ODE) within the variational autoencoder (VAE) framework. Build- ing upon this, Graph Neural ODE (GODE) (such as (Huang et al., 2021) and (Wen et al., 2022)) have been proposed to model the uncertainty in multidimensional time series data and capture its continuous dynamic evolution through latent space. Specifically, these methods integrate the ag- gregation function of GNN as part of the ODE function, effectively capturing the complex interaction information between nodes. This provides an effective tool for modeling incomplete dynamic graph data.
'Key Laboratory of Computational Intelligence and Chinese Information Processing of Ministry of Education, School of Com- puter and Information Technology, Shanxi University, Taiyuan 030006, Shanxi, China. Correspondence to: Jianqing Liang <liangiq ex..ccu.ccn..
However, existing GODE models have limitations in captur- ing the time-varying relationships between nodes and the nonlinear evolution of node states. For example, in trans- portation systems, the road network has significant impact on traffic flow, and its fluctuations are affected by factors
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
such weather, holidays, and emergencies, leading to com- plex nonlinear changes. Therefore, modeling the dynamic relationships and nonlinear evolution of node states remains a key challenge in complex network modeling.
5
4
0
3
2
To address these issues, this paper proposes a model called ControlSynth Graph ODE (CSG-ODE) based on the VAE framework. In the encoding phase, CSG-ODE introduces node importance weights based on information propagation, overcoming the limitations of relying solely on latent space correlations between nodes. By combining node importance with latent correlations, the model effectively guides the learning of temporal node embeddings through an adaptive graph convolutional recurrent network. In the decoding phase, the model uses an ODE to model the dynamic evo- lution of nodes, considering not only the linear changes of nodes and their interactions with other nodes but also capturing the nonlinear evolution characteristics of node states through multiple sub-networks with nonlinear activa- tion functions. This comprehensive approach reflects the dynamic changes in node states. To enhance stability in high-stability scenarios, we introduce the Stable CSG-ODE (SCSG-ODE) model based on CSG-ODE and theoretically demonstrate its improved stability.
T--1 Te
Te+1
Tabs
Figure I. Illustration of five nodes in dynamic graph sequence sample. The left diagram illustrates the original graph structure with fixed connections. The right diagram shows the presence absence of data for each node at each timestamp.
attributes X = VI' Addittonn Addition- ally, we aim to predict node attributes X T-Too for the next F timestamps.
3. Method
In this paper, we propose the CSG-ODE model. Follow- ing the structure of VAE, CSG-ODE consists of two main components: (1) Latent Distribution Generation: Given in- complete dynamic graph data, this component generates an approximate posterior distribution of the latent states for all nodes; (2) Latent State Generation: Based the ODE function we designed, latent states are generated for any given timestamp by sampling the initial state for each node. The overall framework is illustrated in Figure 2.
Our contributions can be summarized in the following three aspects: (1) We propose the CSG-ODE model, which en- ables the model to better capture the complex evolution of node states through an information propagation-bant based inter-node importance weighting method and multiple sub- networks with nonlinear activation functions. (2) Based on CSG-ODE, theoretically give its stabilized version SCSG-ODE. (3) The experimental results show that the pro- posed CSG-ODE outperforms the existing GODE model on five dynamic system datasets. And SCSG-ODE not only outperforms the existing GODE in terms of performance, but also exhibits optimal stability.
3.1. Latent Distribution Generation
In this section, we describe how to learn the approximate posterior distribution of the latent states for all nodes from incomplete dynamic graph data, as shown in Figure 3.
2. Problem Definition
Enhanced Time-varying Relationship between Nodes. Existing GODE models utilize latent representations to capture time-varying node relationships but primarily rely latent space correlations, overlooking the impact of edges on information flow. Inspired by edge importance (Noschese & Reichel, 2024), we propose an Information Propagation- Inter-node Importance Weight to better capture these dynamics. We compute the partial derivative of edge weights with respect to the total communicability of the graph (Benzi & Klymko, 2013). A larger derivative indicates greater contribution to information transmission, making the edge more significant. By combining inter-node importance with latent correlations, our model overcomes the limitations of existing methods that rely solely on latent space- based edge weights, this process is shown in the left part of the Figure 3.
Given a sequence of discrete timestamps T {Ti, T,.... ] E oRobs with non-uniform intervals, we define the corresponding sequence of dynamic graphs gTi.Tase = gr. .To...,TToo }. where the node connec- tivity remains constant and the features vary across times- tamps, as illustrated in Figure 1, At timestamp Tr, the state of the graph is represented as GTe where V is the set of N nodes and È is the set of edges. The subset of nodes with observable features at T₁ is de- noted as V.Z' C V, with feature vectors X', = {x²'
i E VT. RIVT' xd1, Each node has an observation set T C Robs total, Our goal is to learn low-dimensional latent representations Z' E R. xd2 for each node at any timestamp & from the dynamic graph sequence GT..Ta, and use these representations to reconstruct unobserved node
2

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Output
Decoder (xxoerr-p(((z2)
Latent Distribution
Sample
Input
22
Generation
z1
N(H, 1.
Latent State Generation
Encoder
Generative Model and Decoder
Figure 2. An overview of the proposed CSG-ODE.
Specifically, we first represent the latent correlations be- tween nodes using a static adaptive adjacency matrix (Wu al.. 2019). This latent correlation does not require any prior knowledge and can be dynamically adjusted during the model's is learning process, thereby automatically discovering the potential spatial relationships between nodes:
the optimal step size. This choice is based on error analysis of the finite difference method, aiming to balance truncation and rounding errors, thereby minimizing the total numerical error. The notation || || represents the Frobenius norm, with [C ee )[[F corresponding to the total transmis- sibility of the graph. The mixed edge weight between node and node j. denoted as Gigie is computed as follows:
G (ootMMar(ReLU(E..
(1)
where E,, E, E IR Nxd1 represent the learnable embedding of the source and target nodes, respectively, which are ran- domly initialized. By multiplying E, and E,, we obtain the latent correlation matrix G, where G = [G,y] RR XN and Giy represents the edge weight or latent correlation be- tween node and node j. Next, we calculate the importance weights D [Diy] E R N between nodes:
Gmii = Gij + WTJ Dij: Gj
(4)
where W NNN represents the learnable weight matrix. Therefore, by integrating both the inter-node importance and latent correlations, we address the limitations of edge weights that are solely based on latent spatial correlations between nodes. This fusion beneficial for guiding the subsequent learning of temporal node embeddings.
G cc ) x expolGo + +ee )) expolGo. "Bee
Theorem 3.1. For the computation of the node importance weight matrix D E N the time complexity is O(N3).
2,3
(2)
G. G. Ly T _ee (G. eee
TeeT)
Note: Although the subsequent Frobenius norm and associ- ated linear operations incur an additional O(N2) cost, this is negligible in comparison to the dominant (NN)) com- plexity arising from matrix multiplications.
D
(3)
where G, E N represents the original adjacency matrix I,... 1] R. and
of the graph. The vector expo(G.) denotes the matrix exponential minus the identity matrix, represented by the power series expansion of the exponential function (GxPo(G.) = exp(G.) - G. + G
Temporal node embedding. The goal of Latent Distri- bution Generation Moudle is to learn temporal node em- beddings from observable dynamic graph sequences and to approximate the posterior distribution of node latent states based on these embeddings. The temporal node embedding process is shown on the right side of the Figure 3. cap- tures the structural information between nodes by building a graph snapshot at each timestamp. (Zhu al., 2016; Sankar et al., 2020). Inspired by (Luo al., 2024c), we dynami- cally adjust Grii based on sampling density and use mask matrix to construct graph snapshots for guiding the learning of temporal node embeddings. By adjusting edge weights based on temporal sampling density, we allocate node in- fluence according to the differences in sampling density.
G
3f -..).
2f
The term LG T) denotes the Fréchet derivative with respect to G and ee which quantifies the total trans- mission rate and is used to measure the contribution of each edge to the overall information flow in the graph. Equation (2) shows that this quantity is computed using a finite difference approximation, where 3 denotes the step size; smaller values of 3 generally yield higher ap- proximation accuracy. We adopt the empirical formula & 14 (Noschese Reichel, 2024) for determining
3

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Temporal node embedding
Time
Element-wise Addition
Node Embedding Vector
Node-specific Parameters (Graph Convolution)
Projection Heads
Enhanced Time-varying Relationship between Nodes
Go
Information
Propagation-
GiLi Density Dynamic
based
Adjustment
Inter-node
Importance Weight
G
6
Gt
G
Adapt Graph
g
Graph Signal Input
Figure Overview of the Latent Distribution Generation in proposed CSG-ODE.
Detailed information on the dynamic density adjustment mechanism provided in Appendix A. Specifically, the graph snapshot timestamp T, constructed based the dynamic adjustment of time sampling density is as follows:
Due to the varying time evolution patterns different nodes, the latent state distribution of each node also differs. As discussed in (Bai al., 2020), although there may be strong spatial correlations between adjacent nodes, the dynamic nature of time series data and various factors that may in- fluence the nodes (such as node-specific attributes) lead to diverse patterns between different nodes. Inspired by (Bai et al., 2020), we assign dedicated parameters to each node. Specifically. since the number observable nodes varies, further compute the average of all observable value embeddings E, mean E IR for each node and use projection function (pR)):IR* - R29 represented by an M LP to generate each node's 's embedding Q, R2;
C Giji M (1- W alo [
12550
D,
(5) where R.T represents the sampling density of node i at timestamp H, represents the activation function, is the hyperparameter controlling the proportion of the time density, W E NNNN denotes the learnable weight matrix, and M E R N represents the mask matrix at timestamp Tr
1, if both mode and are observed at Tr,
obs obs.total,
M
Emeon
E Tse
(9)
O, otherwise.
obs_total
(6)
(10)
For the graph snapshot at timestamp Tr, we use a Graph Convolutional Recurrent Neural Network (GCRNN) update operator FGCRNN that integrates graph structure and tem- poral dynamics:
The node-specific graph convolution parameters O R Nx are obtained by multiplying the node represen- tation matrix Q E R with weight matrice Wc E Rax/x0 respectively:
T FGCRNN(G". E.. HT-11
+ = QWC
(1I)
where H E Nx is the hidden state at timestamp T, and E NxIXO denotes the graph convolution parameters, with I and O corresponding the input and output feature dimensions, respectively. E. represents the set of all ob- servable embedding vectors at timestamp T,, where each node's s observation is obtained through a M MLPE:
We obtain the approximate posterior distribution of the latent state for each node i by mapping the last observed hidden state H, = H. To using a function Φ(-), and then sample the initial latent state z! E RRda for each node from this approximate posterior distribution:
go (TTTTa)) = -N E,), where p4o = ((H,),
T (T.PE(()))
(12)

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
=>pp(=?) R qo (=][g") TT...
(13)
To address the demands of high-stability scenarios, we pro- pose the Stable CSG-ODE based on the CSG-ODE model.
Detailed information on the construction method of the approximate posterior distribution and the sampling process can be found in the Appendix B.
The SCSG-ODE is characterized by the following ODE form:
M Aʃs (MLP5(2) + +c(), j=1
3.2. Latent State Generation
#
Azf +
(15)
In this section, we describe how to generate the latent state of each node at any given timestamp from its initial state 24. The GODE model captures the evolution of node states over continuous time by solving an ODE, allowing for more accurate modeling of state changes. This provides a key advantage in understanding the evolution of complex dy- namic graphs. Existing methods, such as (Huang et al., 2020; Luo et al., 2024a), typically model node evolution through linear dynamics and interactions with other nodes, neglecting nonlinear state evolution. For example, under adverse weather conditions like heavy rain or snow, traffic flow drops sharply, with sudden congestion emerging and worsening over time. This nonlinearity is often overlooked, limiting the model's ability capture complex dynamics. Inspired by ControlSynth ODE (CSODE) (Mei et al., 2024), we propose a generative model that uses ODE to model both linear and nonlinear node state evolution. By incorporating multiple subnetworks with nonlinear activation functions, our model more effectively captures dynamic changes in node states.
where A is a learnable antisymmetric matrix. Remark 3.2. For any matrix A, we can control it to be an antisymmetric matrix by using A AT.
Consider the ODE defined by Equa- tion (15). Under certain conditions, we can prove that the system is stable. Specifically, when solving this ODE, re- gardless of the initial values (initial states) sampled from the approximate posterior distribution, the state evolution described by the equation remains consistent and is not sig- nificantly affected by the initial values. This demonstrates that our model is insensitive to the initial conditions.
Stability Analysis:
Definition 3.3. The initial conditions are insensitive: Even when the initial conditions of the equation are subject to minor perturbations, the solution will remain within finite and stable range.
When the depth of each subnetwork is set to 1, the subnet- work MLP, can be equivalently represented by weight matrix Wj E Redaxds, Under this setting, we present the stability theorem of Equation (15) as follows:
Specifically, we introduce interactions between nodes control information into the ODE function and use an ad- ditional control function to represent the external influence on the nodes. At the same time, we employ multiple sub- networks with nonlinear activation functions to capture the nonlinear changes in node states:
Theorem 3.4. Assuming that the control information di does not depend on 2/, and that for any i,j E {1,2, M) the condition W,W, W.W, holds. Then, the Equa- tion (15) is stable.
M AsS (MIP) + ((4),,
# A0z
Σ
Based Theorem 3.4, of the system ensures that when differrnt initial latent states of are sampled from the approximate posterior distribution, the solution of Equation (15) will not experience drastic fluctuations. Even in the presence of small perturbations, the solution remains stable within a finite range, preventing significant oscillations. Theilami 3.4 provides theoretical guarantees for the stability of the model. A detailed proof of this theorem can be found in the Appendix C.
conclude that the stability
(14)
& = (2N 25, x
In this model, oi represents the interactions between nodes, i... the external control information of the nodes obtained through a GNN (Kipf et al., 2018). The control information directly influences the latent state of the nodes through control function g(-). wh-ch is parameterized by a neural network. This mechanism helps the nodes to adapt to ex- ternal factors, thereby introducing dynamic adjustments to node behavior within the model. We denote the subnetwork as MLP5. J. with f-) representing the activation function used by subnetwork j (such as tanh or ReLU, etc.). In this work, we maintain that each subnetwork uses the same acti- vation function (ʃi = f2
By solving the ODE function, we obtain the latent state at the desired time step:
pred
# .... 2
O TAEESolveEEquation (14), .., 28], (lo, ... bpred) ).
SMr). The matrix Aj (for j = 1,2,.... M) represents the Iearnable weight matrix, which ensures that the weight matrices appropriately com- bine to form the overall output of the ODE, thus enhancing the model's expressive capacity
(16)
Given the initial latent states of each node 20, 22 N analytical solutions for ODE are typically unavailable. Therefore, numerical methods, such as Euler's method, are employed to approximate the solution, A numerical ODE
0 0
5

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
solver can compute the approximate value of the node state of any time step i.assisting in simulating the system's behavior over time.
uate model performance using the Mean Squared Error (MSE) metric (Huang et al., 2020; Luo et al., 2024a; Huang et al., 2024), comparing the reconstructed tra- jectory with the true trajectory.
Finally, a decoder, represented by a neural network, is used to decode the probability p(a] and recover the node features . Under the VAE framework, all modules are jointly trained by maximizing the evidence lower bound (ELBO) of the likelihood function, as shown below:
We split the time range into two segments: (40, (m) and (n). During training, the model reconstructs the trajectory for the second seg- ment (fx1, 4n) using observations from the first seg- ment. In testing. the model predicts the trajectory for the future time range (In, ln2) based on observations from (to, b. The model is conditioned on partial ob- servations (40%, 60%, and 80%) from the first segment, and the reconstructed trajectory is evaluated using the MSE metric.
Extrapolation Task:
LELBO- E est=:1o7h [10gp(2 [
KL
go (2) gth (1 )[p(Zz))
(17)
where p(Z)) = IT= p(z)) and p(=?) is a standard normal distribution. The training process of this model is detailed in Algorithm 1 in the Appendix I.
Further details on the experimental setup and hyperparame- ters can be found in Appendix E.
4. Experiments
4.4. Results
4.1. Datasets
Table presents the MSE for interpolation tasks across dif- ferent datasets and methods. Latent-ODE, which does not consider the interaction information between nodes, exhibits poor performance. Edge-GNN, although propagating tem- poral dynamic information in the graph, fails to capture complex temporal dependencies, resulting in suboptimal performance. Weight-Decay does not fully account for the dynamic interactions between nodes and only performs a simple parameterization of time intervals, leading to unsat- isfactory results. LG-ODE leverages neural ODE to infer complex latent dynamics but still falls short in modeling the temporal relationships and nonlinear evolution between nodes, limiting its performance. Although LG-CSODE em- ploys our designed ODE function for modeling, it relies solely on latent correlations model node relationships dur- ing encoding, which negatively impacts performance. As the observation ratio increases, the reconstruction loss for all models decreases, as expected, because more observed data helps the model more accurately reconstruct the unobserved portions.
We evaluate our model on five diverse datasets: two syn- thetic datasets, Springs (Kipf et al., 2018) and Charged (Kipf et al., 2018), and three real-world datasets- CMU motion capture data (walk capture from subject 35 and jump capture from subject 1 118) (CMU, 2003), and the PEMS08 traffic flow dataset (Song et al., 2020). Further details are provided in Appendix D.
4.2. Baselines
To evaluate the effectiveness of the proposed CSG-ODE in the task, we designed comparative study with several exist- ing methods, including four baseline models: Latent-ODE (Rubanova et al., 2019), Edge-GNN (Gong & Cheng, 2019), Weight-Decay (Cao et al., 20, and LG-ODE (Huang et al., 2020), as well as two advanced model variants: NRI+RNN (Huang et al., 2020) and LG-CSODE. Detailed descriptions of these models can be found in the Appendix F.
4.3. Experiment Setup
Table 2 shows the MSE for extrapolation tasks, which is higher compared to interpolation tasks, as predicting future trajectories is more challenging than reconstructing missing data. Similar to interpolation tasks, the prediction errors for all models decrease gradually as the observation ratio increases. Our model demonstrates significant performance improvements across multiple datasets and settings, further validating the effectiveness of the proposed design. The results are visualized in the Appendix G.
We evaluate model on two tasks(Rubanova et al., 2019):
Given a dynamic graph with evolving nodes over the time range (lo, (n), we ob- serve their state features at specific time points, with partial observations due to practical constraints. The task is to predict the complete dynamic trajectory of nodes over the time range {4o, .,,,,,,,,,, J. based on a subset of observed data. The subsampling ratio is set to 40% 60%, and 80%, and the observations are independent across nodes. The objective is to predict the missing feature values for all time points. We eval-
Interpolation Task:
Furthermore, we analyzed the potential sources of error in CSG-ODE. The model assumes that the underlying interac- tion graph remains static over time; however, this assump- tion does not fully hold in real physical systems, which may
6

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Table 1. Mean Squared Error (MSEx10 on Interpolation task.
Dataset
Springs
Charged
Motion-walk
Motion-jump
PEMS08
ration
40y
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
ADtent-ODE
0.4729
0.4207
0.3757
1.0185
1.0352
0.7411
0.1405
0.0799
0.0766
0.1598
0.1688
0.1028
0.4855
0.4710
0.4672
Weight-Decay
0.9127
0,9504
0.9416
2.4532
2.0855
1.3690
0,3465
0.3314
0.3139
0,3940
0,7005
0,4820
1.0356
1.0641
1.1443
Edge-GNN
1,0488
1.0681
0.7171
1.3635
1.4446
1.0158
0.5044
0.4414
0.4305
0.5736
0.5571
0,5274
1.1901
1.1959
1.0826
NRI-RNN
0.4098
0.3382
0.3107
1.2010
1.0784
0.9201
0.1065
0.0893
0.1211
0.1212
0.1287
0.1185
0.4651
0.3787
0.4691
LG-ODE
0.2628
0.2648
0.2313
0.7971
0.7656
0.7157
0.0823
0.0475
0.0795
0.0936
0,1004
0.0974
0.2982
0.2965
0.3492
LG LG-CSODE
0.2155
0.1666
0.1937
0.7794
0.7556
0.7229
0.0461
0.0426
0.0435
0.0665
0.0648
0.0649
0.3214
0,2890
0.3525
Ours
0.1550
0.1440
0.1386
0.7947
0,7169
0.7099
0,0439
0,0406
0,0400
0.0426
0,0414
0,0405
0.2526
0.2827
0_3360
Table 2. Mean Squared Error (MSEx 10 3) Extrapolation task.
Dataset
Springs
Charged
Motion walk
Motion Jotiion-uup
PEMS08
ration
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
Latent-ODE
5.5751
3,7938
3.8878
11.7554
10,7649
18.8735
0.7222
1.0262
0.9042
0,5166
0.5331
0.5947
6.2301
4.4778
3,8409
Weight-Decay
5.1282
5.1279
4.8347
8.2000
7.6614
7.3471
5.0175
4.8270
4.6517
3,5887
2,9780
3.0594
5.7307
6.0525
4.7664
Edge-GNN
5.0331
4.3959
2.9056
7.9716
7.6917
7,1133
3,9712
4,9322
3,3552
2,8404
2.7429
2.2067
5.6244
5,1885
3.8706
NRI+RNN
2.2191
2.1437
2.2998
6,2108
5,8829
6.1291
1.0565
1.0636
0.8843
0.7556
0.6562
0.5816
2.4798
2.5302
2.2721
LG-ODE
1.4861
1.6151
1,5427
5.6522
5.4162
5,7353
0.3835
0.4391
0.4084
0,2743
0.2709
0.2686
1.7726
1.9063
1.5241
LG CSODE
1.3508
1.2975
1,2750
5.8291
5.5811
5.6171
0,1853
0.1583
0.1638
0,2536
0.2837
0,3242
2.4391
3.5231
3.4111
Ours 1.3495
1,2969
1,2691
5.5086
4.7690
4.4966
0.1791
0.1539
0.1593
0.1393
0.1290
0.1248
1.6607
1.7436
1.4937
introduce a certain degree of bias. Numerical errors inherent in the ODE solving process are also unavoidable. In addi- tion, real-world data are often influenced by complex factors such as environmental changes, noise, and latent unmod- eled dynamies, none of which are considered in the current model, potentially further exacerbating error accumulation.
node representations yields better interpretability and per- formance than adaptive node embeddings. (2) Removing the information transmission buasd inter-node importance weighting causes a significant performance drop, as this weighting encodes both the graph topology and the role of edges in global information propagation. Integrating this weighting with latent pairwise correlations enhances the model's ability to capture time-varying inter-node rela- tionships. (3) Excluding the control function g(-) degrades performance, since g(-) bridges external control inputs and internal dynamics; its absence reduces the node's sensitivity to external influences. (4) Removing the nonlinear term weakens the model's capacity capture complex dynamics, as real-world systems are inherently nonlinear; incorporat- ing nonlinearity improves expressiveness and adaptability.
4.5. Ablation Study
To further analyze the components of the model, we conduct an ablation study by considering four model variants. First, due to the differing time patterns across nodes, simply shar- ing a single parameter space for all nodes is insufficient to capture these differentiated time dependencies. Therefore, we adjust the parameter space for each node according to its representation to accommodate its respective time pattern. We compare the learnable adaptive node representation ma- trix Q E Vxg with the handcrafted node representation matrix, where the former is denoted as Ours-AQ. Addi- tionally, remove the node importance weights based information propagation, resulting in the variant Ours-no EL. Second, in terms of the control term in the neural differ- ential equation, we remove the control function g(-), which influences the node states, effectively using only the interac- tion information between nodes. This variant denoted as Ours-no g()). Finally, we remove the nonlinear term in the neural ODE equation, retaining only the control informa- tion and linear terms; this variant is referred to as Ours-no NI. Through these variants, we aim to gain a deeper under- standing of the role of each component in the model and its contribution to the overall performance.
4.6. Parameter Sensitivity
To investigate the impact of the hyperparameter a, which controls the proportion of time density, we conducted experi- ments using walk capture data from Subject 35. We varied from the set {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1} and analyzed its effect across two tasks. As shown in Fig- ure 4, for the interpolation task, changes in a had minimal impact, and the model's performance remained stable. This suggests that, regardless of sampling density, the model can effectivvly infer missing data based on existing trends. For the extrapolation task, we observed that when the observa- tion ratio was low, changes in a had little effect. However, as the observation ratio increased, performance improved with higher sampling density, as the increased density bet- ter captured the data dependencies. Accordingly, we se- lected = 0.5 for the majority of experiments, since the
The results in Table 3 and Table 4 reveal several key find- ings: (1) Using the mean of node observations to construct
7

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Table 3. Ablation study on all datasets (MSE x x10 3) for the Interpolation task.
Dalaset
Springs
Charged
Motion walk
Motion-jumpp
PEMSO8
ration
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
CSG-ODE AQ
0.1520
0.1526
0.1465
1.8275
1.0288
0.8397
0.0674
0.0614
0.0618
0.0880
0.0691
0.0664
0.7949
0.6989
0.6058
CSG
0.1841
0.1712
0.1577
0,8005
0.7325
0.7855
0.0470
0.0435
0.0411
0.0440
0.0429
0.0413
0.2644
0.2531
0.4137
CSG-ODE no x()
0.1655
0.1918
0.1417
0.8236
0.6531
0.7422
0.0480
0.0440
0.0433
0,0432
0.0449
0.0436
0.2595
0.2466
0.4857
CSG LL-O no Ni
0.1913
0.1592
0.1507
0.7950
0.7775
0.7768
0.0456
0.0426
0.0478
0.0430
0.0409
0,0443
0.2681
0.2811
0.3744
CSG-ODE
0.1550
0.1440
0.1386
0.7947
0.7169
0.7099
0.0439
0.0406
0.0400
0.0426
0.0414
0.0405
0.2526
0,2602
0.3360
Table 4. Ablation study on all datasets (MSE x x10-3 for the Extrapolation task.
Dataset
Springs
Charged
Mation walk
Motion Motion-jump
PEMS08
ration
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
40%
60%
80%
CSG ODE AQ
4,4046
3.0096
2.0561
11.8165
8.3525
6.5031
0.2337
0.1840
0.1695
0.2620
0.2075
0.1978
2.8898
2,6808
2.5539
CSG-ODE
2.4394
2.0790
16635
5.6443
5,0040
4.5996
0.1784
0.1549
0.1212
0.1413
0.1298
0.1253
1.9424
1.7881
1.7049
CSG-ODE s(-)
2.4239
2.0039
1.4903
5.6228
5.0025
4.6865
0.1849
0.1545
0.1494
0.1424
0.1311
0.1264
1.9448
1.6178
2.2629
CSG ODE no NI
1.5368
1,5077
1.4983
5,5850
5.0550
4.5048
0,1954
0.1635
0.1601
0,1405
0.1293
0.1261
1.7763
1.8778
1,5109
CSG-ODE
1.3495
1.2969
1,2691
5,5086
4.7690
4,4966
0.1791
0.1539
0.1593
0.1393
0.1290
0.1248
1.7726
1.7436
1.4937
results tend to stabilize and achieve a local optimum around a= = 0.5 in the extrapolation experiments.
Table S. Performance comparison of SCSG-ODE, CSG-ODE, LG- ODE, NRI+RNN, Edge-GNN
Seeet- and Latent-ODe motion-walk datasets. The best and second best results are highlighted in blue and brown, respectively.
4.7. SCSG-ODE Performance
We performed comparison experiment of SCSG-ODE with walk capture data from subject 35 with baselines. The Ta- ble 5 indicate that CSG-ODE outperforms all other models, while SCSG-ODE achieves near-optimal performance, a result that reasonable. Compared to CSG-ODE, SCSG- ODE fixes the learnable weight matrix to the same skew- symmetric matrix, while different subnetworks simulate the nonlinear evolution of the nodes. Using distinct weight matrices ensures that these subnetworks, when appropri- ately combined, produce the overall output of the model, thus enhancing the model's expressive power. Although CSG-ODE outperforms SCSG-ODE, we have theoretically demonstrated that by controlling the weight matrices, this design significantly improves the model's stability. More- over, relative to LG-ODE, NRI+RNN, Edge-GNN, Weight- Decay, and Latent-ODE models, the performance of SCSG- ODE is optimal. To verify the correctness of the theoretical derivations, we conducted additional experiments on the per- formance of CSG-ODE and SCSG-ODE in the extrapolation task. Specifically, we conducted 5 rounds of experiments for each model with different subsampling ratios walk motion capture data from subject 35 and calculated their means and standard deviations. The experimental results are shown in the Table 6. It is worth noting that the stan- dard deviation of SCSG-ODE is always about half of that of CSG-ODE under all subsampling ratios, which indicates that SCSG-ODE significantly improves the stability of the model under the same experimental setup, and thus verifies the correctness of our theoretical derivations. Therefore, the SCSG-ODE is an ideal choice when we face the demands of scenarios that require high stability.
Task
Interpolation
Extrapolation
ratio
40%-
60%
80%
40%
60%
80%
Latent-ODE
0.1405
0.0799
0.0766
0.7222
1.0262
0.9042
Deggtt-Decay
0.3465
0.3314
0.3139
5.0175
4.8270
4.6517
Edge-GiN
0.5044
01.4114
0.4305
3.9712
4,9322
3.3552
NRI-RNN
0.1065
0.0893
0.1211
1.0565
1.0636
0.8843
LG-ODE
0.0823
0.0475
0.0795
0.3835
0,4391
0,4084
CSG-ODE
0.0439
0.0406
0,0400
0.1791
0.1539
0.1593
SCSG-ODE
0.0435
0.0410
0,0438
0.2447
0,2035
0.1883
Table 6. Comparison of the stability performance of SCSG-ODE and CSG-ODE motion-walk.
ratio CSG-ODE SCSG-ODE
40%
60%
80%
0. 0.1883 + 0.0092
0,1676 + 0.0109
0,1524 + 0.0097
0.2304 + 0.0056
0.1978 + 0.0050
0.1787 + 0.0043
5. Related Work
5.1. GNN for Dynamic Graph Representation
GNN (Scarselli et al., 2008) have become a powerful tool for dynamic graph representation learning. In dynamic graphs, time affects not only node features and edge weights but also the structure of the graph itself. Recent GNN-based models (Wu et al., 2019; Hajiramezanali et al., 2019; Rossi et al., 2020) focus on capturing temporal changes in node features. These models have been widely applied in areas such as traffic prediction (Zhao al., 2019; Guo et al., 2020) and healthcare (Gao et al., 2021; Wang & Jin, 2025). ever, most of these models assume complete, synchronized observations with uniform time intervals. This assumption limits their effectiveness in handling incomplete data, par- ticularly when time intervals are irregular. To address this,
8

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Extrapolation Task
Interpolation Task
0.19
0.016
605
0.18
0.045
025
00%
90%
M
ASSE<x10
0.16
0.
0.041
0.010
0.039
0.
0.038
Figure 4. Comparison of the effect of different values model performance in two tasks.
we introduce VAB-based framework, using GNN en- coder to infer an approximate posterior distribution of node latent states from irregularly sampled partial data. We also integrate a neural ODE to model the dynamic evolution of node states, enabling the generation of latent node states at any timestamp and overcoming the limitations of existing methods in incomplete data scenarios.
tions between nodes but overlook nonlinear changes in node states. In contrast, our model addresses this limitation by introducing node importance weights based on information propagation. During the decoding phase, we use multiple subnetworks with nonlinear activation functions to capture the nonlinear evolution of node states, providing a more comprehensive representation of their dynamic behavior.
5.2. NODE for Irregularly Sampled Multivariate Time Series Modeling
6. Conclusion
The CSG-ODE model proposed in this paper effectively addresses the limitations existing GODE models in mod- eling time-varying relationships between nodes by introduc- ing node importance weights based information propaga- tion. Additionally, the model utilizes multiple sub-networks with nonlinear activation functions to capture the nonlinear evolution of node states, further enhancing its expressive power. To improve the model's 's stability, we introduce an ex- tended version, SCSG-ODE, and thcoretically demonstrate that this extension significantly enhances stability. Exper- imental results show that CSG-ODE outperforms existing GODE models across multiple datasets. Although CSG- ODE outperforms SCSG-ODE in terms of performance, the latter demonstrates a more pronounced advantage in sta- bility. Future research could focus on further optimizing the computational efficiency and generalization ability of both CSG-ODE and SCSG-ODE for large-scale and more complex graph data.
Modeling irregularly sampled multivariate time series is challenging due uneven time intervals and incomplete ob- servations. Neural Ordinary Differential Equations (NODE) (Chen et al., 2018) combine neural networks and ODE to model dynamic systems and have proven effective for time series data. Several variants have been proposed (Rubanova al., 2019; Kidger al., 2020; Mei et al., 2024), with some addressing the GNN over-smoothing problem and extending its use in multivariate time series modeling through GNN or message-passing mechanisms (Deng et al., 2019; Poli et al., 2019; Xhonneux et al., Recently GODE has been in- tegrated into the VAE framework to model irregularly sam- pled multivariate time series (Huang et al., 2021; Wen et al., 2022; Yıldız et al., 2022: Luo al., 2024b; Gravina al., 2024). LG-ODE (Huang al., 2020) employs unsupervised learning to infer initial states from irregular observations and uses NODE for continuous-time latent dynamics. (Yildiz et al., 2022) accurately decompose the independent dynam- ics of individual objects from their interactions and infer the independent dynamics and its interaction with reliable uncertainty estimates using ODE for underlying Gaussian processes. PGODE (Luo et al., 2024a) incorporates latent representations into the graph ODE, enhancing the model's expressiveness by identifying interaction prototypes. How- ever, existing GODE models mainly capture latent correla-
Acknowledgement
This work is supported by the National Natural Sci- ence Foundation of China (Nos. 62272285, 62376142, U21A20473).

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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Impact Statement
Gravina, A., Lovisotto, G., Gallicchio, C., Bacciu, D., and Grohnfeldt, C. Long range propagation on continuous- time dynamic graphs. In Proceedings of the 41st Inter- national Conference on Machine Learning (ICML 2024), pp. 16206-16225. PMLR, 2024.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Guo, K. Hu, Y., Qian, Z., Sun, Y., Gao. J., and Yin, B. Dynamic graph convolution network for traffic forecast- ing based on latent network of laplace matrix estimation. IEEE Transactions on Intelligent Transportation Systems, 23(2):1009-1018, 2020.
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Knowledge Discovery and Data Mining (KDD 2023). pp- 3650-3662. Association for Computing Machinery, 2023.
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
A. Detail of Density-aware Dynamic Adjustment
By adjusting the edge weights between nodes based on temporal sampling density (Luo et al., 2024c), we can reasonably allocate the influence of each node in the graph according to the differences in sampling density. Nodes with lower sampling density may have less accurate or incomplete information, so their edge weights with other nodes are reduced to avoid excessive interference from inaccurate state data during node evolution.In contrast, nodes with higher sampling density have more accurate information, enabling effective information sharing with other nodes, thus their edge weights are larger, ensuring their influence in the graph. The calculation formula for the sampling density R of node i at each observable timestamp T3 is as follows:
if both Ti-1. and Ti+1, exist, if only T--1, exist, if only Ti+1, exist, To if neither Ti-1, nor T4+1. exist. 2
Te, T--1a
R
T++.4 - Th,
The sampling density is determined by calculating the average time interval between each observation point and its preceding and succeeding observation points. For timestamp Te, if node i has no preceding succeeding observation points at this timestamp, it indicates that this is the only observation point for node i. In this case, its sampling density is set to half of the maximum observation time span.
B. Detail of Distribution Generation and Sampling
To obtain the approximate posterior distribution of the latent state for each node, input the hidden state H; R updated at the last observed timestamp of node into MLP ((-): Re + R22a, which outputs a distribution vector HD E R22d2;
H" D (H,).
Next, we split this vector into two parts, with the first d2 dg-dimensional vector representing the mean vector #4 (4,,,,44,,..., ..... Pi.
d2 of node i, and the second dy-dimensional vector representing the variance vector Ø; [0r,1, oi,,, Fi,dal E da of node i. Notably, the approximate posterior distribution of the latent state generated by the node is a da-dimensional Gaussian distribution N (ps, E.), with the assumption that the distributions across dimensions are independent. Thus, the covariance matrix Ejis represented as a diagonal matrix diag(a oz,; .... os,d)), where the diagonal elements correspond to the variance in each dimension of the variance vector, and each element of μi; and o; corresponds to the mean and variance of the Gaussian distribution in each dimension, respectively.
We apply the Reparameterization Trick (Kingma et al., 2013) to sample the initial latent state 2 E 12 for each node from this approximate posterior distribution (u,, E.). Specifically, when sampling from multivariate Gaussian distribution where each dimension's 's distribution is independent, we first sample from each univariate Gaussian distribution, then combine them into a dz-dimensional vector:
2 (Mi1 + 041 * #1 M1, 0i, * E2, Mi,da + os Edz),
where 5 (O, 1) is sampled from the standard normal distribution.
C. Proof of Theorem 3.4
Lemma C.1. (Haber Ruthotto, 2017; Gravina et al., 2024)The equation stable if and only if the real part of all eigenvalues of the Jacobian matrix (1) of the equation is zero, i.e., Re():(J(I)) -l.
Proof. Since we set the depth of each subnetwork to 1 and the subnetwork MLP, is represented as the weight matrix W,, the equation can be simplified to the following form:
$ = A2t Af)(W(2!)) + ((()))
The eigenvalues of the Jacobian matrix are key factors controlling the sensitivity of the solution to initial conditions. The Jacobian matrix describes how the system state responds to small perturbations, and its eigenvalues determine the rate of
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
expansion contraction of the system in different directions.
J(t) (T+ Σ (
Let P P (z)' the space of non-diagonal real d2 x d2 matrices, the set of non-diagonalizable matrices (in the complex field) has zero measure (Bhatia, 2013). Therefore, under random initialization, the matrix W, is diagonalizable with probability Lie., W VyAsV;",, where A, = diag(>).).... ),,)), E Cd2xd2, By assumption, for any i, e .
{1, 2,., M), all W,W; = W,W holds, then all matrices Wj can be diagonalized simultaneously. Hence, by similarity, we can rewrite as:
P= (T+ (+)((()))
Moreover, since the derivative of the activation function tanh is non-negenn non-negative, P is clearly invertible. Therefore, the Jacobian matrix is the result of the matrix multiplication between invertible diagonal matrix and the weight matrix, i.e.,
J (t) = PA.
Let X be eigenvalue of PA with the corresponding eigenvector U. We haves
PAT = Xo,
AT = AP-15,
1 AT A (T'P-17)
where denotes the conjugate transpose, and is important note that 7 P- is real number. Since A is anti-symmetric matrix over the complex field, we have A A AA,add thus
(0 A7) = 7 - A..
This implies that T" AT must be purely imaginary, because the conjugate transpose of real number equals its negative, which only holds when the number is purely imaginary. Since र Au is purely imaginary, it follows that must be purely imaginary, i.e, (J())((())) = for all i 1,...,dd,, d2. By Lemma C. this proves that the equation is stable.
D. Dataset Details
D.I. Simulated Datasets
Following the method in (Kipf et al., 2018), constructed two simulated datasets: Spring and Charged. In the spring system, particles are interconnected by springs, and their interactions follow Hooke's law, Each particle interacts with others with equal probability. In the charged particle system, particles carry electric charges and interact via electromagnetic forces, which can be either attractive or repulsive, each occurring with equal probability. Each dataset comprises samples of five particles evolving within 2D box without external forces (aside from possible wall collisions). Particle trajectories are simulated by solving two distinct partial differential equations (PDEs), one for each interaction type. Simulations run for 6000 time steps, with data subsampled every 100 steps. Each particle is represented by four features: position and velocity in both and y directions. To simulate irregular partial observations, For each particle, the number of observed points n. is drawn from uniform distribution U40, 52), and time steps are uniformly sampled from the 6000-step trajectory for training. For extrapolation evaluation, 40 observations are sampled from steps [6000, 12000] using the same method. Sampling is performed independently for each particle. Following (Huang et al., 2020), we generate 20k training and 5k test samples for both datasets. All features are standardized to have a maximum absolute value of 1 in both the training and test sets.
adopt the approach from (Huang al., 2020).
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Table 7. Statistics of the datasets used in the experiments.
Spring
Charged
Motion-walk
Motion-jump
PEMS08
#Nodes
5
29
29
170
#Node ft
4
4
6
3
#Edges
1-20
25
56
56
548
#Edge ft
2
#Train
20000
20000
16
21
199
#Test
5000
5000
49
#Number of samples in Train
U(40,52)
U(40,52)
U(30,42)
U(30,42)
U(40,52)
#Number of samples in Test
U(40,52)+40
U(40,52)+40
U(30,42)+40
U(30,42)400
U(40,52)+40
D.2. CMU Datasets
The CMU dataset (CMU, 2003) is used for human pose recognition and motion analysis, containing thrce-dimensional human motion data obtained through motion capture systems such as optical cameras or sensors. We selected two datasets: the walk sequence of subject 35 and the jump sequence of subject 118. Each sample consists of 29 trajectories, each tracking joint. Each joint has six features: displacement along the x, y, and axes, and angles relative to the x, y, and axes. Similar to the simulated datasets, for each joint in the motion capture data, sample the number of observations from uniform distribution U (300, 42). and uniformly extract n observations from the first 50 frames as the training trajectory. For testing, 40 observations are sampled from frames [51, 15,. We divide the walk experiments into non-overlapping training (16 trials) and test sets (7 trials), and similarly, the jump experiments into -oeeoverlapiing training (21 trials) and test sets (9 trials). All features are normalized such that the maximum absolute value in both the training and testing datasets.
D.3. PEMS08
The PEMSO8 dataset (Song et al., 2020) originates from the traffic flow monitoring system in California, USA and records traffic flow data from various sensor points over different time periods. The PEMS08 dataset includes traffic flow data from 170 nodes over a continuous 62-day period, starting from July 1,2016. The data collection frequency every 5 minutes, with each sample containing three features: the number of vehicles passing through in 5 minutes, the speed of the vehicles, and the occupancy rate. Similar to the generated datasets, for the training data, we divide the first 1,,940 time steps into groups of 60 time steps, resulting in 199 non- con-overlappinng subsequences. Observations for each node are randomly sampled from a uniform distribution U U(4,, (40, 52), with n observations selected from each group of 60 time steps. For the test set, data from time steps [1 11,941,17,8221 is selected, and every 120 time steps are divided into group, generating 49 non-overlapping subsequences. For each test sequence, the first 60 time steps are sampled using the same method as the training set, and 40 observations are randomly selected from the 61st the 120th time steps. All features are standardized, ensuring that the maximum absolute value in both the training and test datasets is 1.
The statistics of the five datasets are reported in Table 7.
E. Experiment Setup
F.1. Interpolation
Fora system consisting of multiple nodes, the first observable time point for each node may differ. This means that the observation timestamps for each node are not aligned, and thus we cannot simply use a fixed time point as the starting time for all nodes. To handle these misaligned observation times consistently, we define a unified starting time fstart 0 for all nodes, which serves as the initial time point for the system, and normalize all observation times to the range [0. 1]. This normalization process ensures that the time series for all objects are processed within unified time frame, eliminating the issue of inconsistent time points across objects. By applying such time normalization, we assume that interactions between objects are continuous, starting from Istan. = 0 and modeled within the time range [0,1]. This allows us to apply ODE within this standardized time frame to describe the dynamic behavior of the system and infer the complete traiectories based on the qiven partial observation data.
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Table 8. Hyperparameter settings for every dataset.
Hyper parameter
Values
Spring 256
Charged
Motion-walk
Motion-jump
PEMS08
batch learing rate dropout k
256
8
32
4
0.0005
0.2
64
64
64
64
16
q
32
32
32
32
Alpha
0.5
M
2
clip epoch h
10
50
16
augment_diin dim I.2
64
64
64
64
0
0.001
subnetworks. width
128
128
128
128
64
subnetworks ODFsolve
Fuler
Optimizer
Adam
E.2. Extrapolation
During the training process, we manually segment each sequence, selecting the system's 's starting time as Astart 4, 2 ,nd the observations prior to lstart are fed into the encoder to estimate the latent initial state. Observations at or after tstart are treated as ground truth data for reconstructing the latter part of the trajectory. For the testing process, in addition to using the first half of the data from (fo, we also sample observation data from the range (t., fn2) evaluate the model's 's extrapolation capability. Therefore, we set start Lw' with observations before in being input into the encoder and observations or after fn used ground truth data reconstruct the latter part of the trajectory. All observation times normalized to the range [0, 1].
E.3. Plyper-Paameee
In the Table 8, report the hyperparameters used for all datasets in the experiments.
F. Details of Baselines
We first consider Latent-ODE (Rubanova et al., 2019), which suitable for modeling a single incomplete time series. This method uses NODE to encode information from incompletely sampled time points, but it does not account for interactions between nodes, making incapable of directly capturing information transfer between nodes. Edge- GNN (Gong & Cheng, 2019) models time intervals as edge attributes, combining temporal information with graph structure to learn interactions between nodes via GNN. The time intervals are reflected in the edge attributes through a weighting mechanism, thus propagating temporal dynamic information across the graph. Weight-Decay (Cao et al., 2018) employs a simple exponential decay function to model time intervals as h(t + At) exp( - h(tt.cap the influence of time intervals on features through the decay function. However, it does not fully consider the dynamid interactions between nodes, as is merely a simple parametrization of time. RNN-D (Che et al., 2018) is a multivariate time series (MTS) model based on RNNs for handling incomplete data. fills missing values by connecting the feature vectors of all nodes uniformly, but neglects interactions between nodes and assumes independence across all nodes, thus failing to model dynamic interactions and capture information transfer between nodes. NRI (Kipf et al., 2018) is method for modeling relationships between nodes and predicting time series in dynamic systems, particularly suited for data with a graph structure. (Huang et al., 2020) combines RNN-D with NRI to form RNN-NRI, which is evaluated in two tasks. LG-ODE (Huang et al., 2020) combines the interaction functions of GNN with NODE, providing a flexible and powerful tool for modeling dynamic systems. It excels in continuous- time dynamic modeling, handling irregular sampling, and utilizing node interactions, making it suitable for a wide range of dynamic graph tasks. Additionally, we replace the ODE function in LG-ODE with the ODE function we
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
designed, retaining the encoding scheme of LG-ODE. This modelis referred LG-CSODE, LG-CSODE is better suited for complex node evolution, offering significant advantages in capturing the nonlinear evolution of node states.
G. Results Visualization
The Figure 5 visualizes the results of our model in the interpolation task on the Spring dataset, with observation percentages of 40% and 60%. The Figure illustrates the results of our model in the extrapolation task on the Spring dataset
Groundtruth
Observed (40%)
Interpolation (40%)
Observed (60%)
Interpolation (60%)
Figure 5. Visualization of interpolation results for spring system.
Groundtruth
Extrapolation (40%)
Extrapolation (60%)
Extrapolation (80%)
Figure Visualization of extrapolation results for spring system.
H. More Experiment Results
H.I. Model Scaling
In the previous experiments, our model demonstrated significant advantages over various baseline models while maintaining the same number of parameters and architecture configuration. Building these findings, extended experiments primarily focus on exploring the scalability of the model, rather than further comparisons with other models Taking the walk capture data from Subject 35 as an example, we investigate the impact of increasing the number of sub-networks, sub-network width, and sub-network depth on system performance. Specifically, the network width refers to the number of hidden layers in the fully connected layers, set to 128, 256, 512, 1024, and 2048; the network depth refers to the number of hidden layers the fully connected layers, set to 1,2,3,4,and 5; and the number of sub- nebboeessrks is 1,2,3,4,am 4,nd
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Interpolation Task
Interpolation Task
Interpolation Task
6.000
0.000
044
4.911
FFA
1.042
128
MMSB(x10
5,962
MSB
2.080
4.438
0,108
0.004
5.039
121
352
512
1224
2041
Numbers sub-networks
Depth
Extrapolation Task
Extrapolation Task
Extrapolation Task
0.19
0.10
0.18
0.10
1.
125
0.11
271.1
12%
0.17
MSB
455
3.19
0.14
0.13
0.12
Numberrs sub-networks
Width
Depth
Figure 7. Comparison of the performance of the CSG-ODE model that increases only the number of sub-networks, sub-network width, sub-network depth in both tasks (with fixed learning rate).
To maintain consistency across experiments, when adjusting one variable, the other two variables and the model's other hyperparameters (e.g., learning rate) remain unchanged.
In this experiment, we controlled the learning rate to be 0.0005, and the experimental results are shown in the Figure 7. In terms of overall performance, increasing the network width network depth, and number of sub-networks resulted in improved performance for both tasks. Moreover, we found that the MSE for both tasks first decreased and then increased as the three parameters above were increased, with the minimum MSE typically occurring whem these parameters were adjusted their median values. For the CSODE (Mei al., 2024) model, the learning rate formula used the design of this experiment is: learning rate width /M VM* where & is constant, and M is the number of sub-networks. This formula adjusts the learning rate based on the width and moderately adjusts it according to the number sub- sub-networks to address the complexity. For example, with width of 512 and sub-networks, the learning rate is 512xvI' Therefore, we applied this learning rate formula during the validation process to adjust the initial learning rate and observe the model's performance as complexity increases. Using the same data, we conducted experiments for both tasks with 60% observation ratio, and the experimental results are shown in the Figure 8. The results indicate that increasing the network width, the number of sub-networks, and appropriately adjusting the learning rate lead stable performance improvement for our model.
H.2. Impact of Solver Choice
Table 9, Comparison of Mcan Squared Error (MSBx 10 -0 -3)
and running time of Euler and Dopri5 solvers for two tasks
Interpolation
Extrapolation
40%
Time
60%
Time
80%
Time
40%
Time
60%
Time
80%
Time
Dopri5
0.0427
1600s
0.0384
1700s
0.0377
1800s
0.1583
900s
0.0919
900s
0.0972
1000s
Euler
0.0439
900s
0.0406
950s
0.0400
1050s
0.1791
550s
0.1539
550s
0.1593
600s
Differential equations are fundamental tools for describing many dynamic systems, and NODE use these equations model
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
Interpolation Task
Extrapolation Task
0.12
0.11
B.1549
2.1055
0.1087
0.2363
0.1450
6.36
256
B0443
3.0161
0.2470
0.0300
0/03355
0.09
256
0.1267
0.1300
0.1342
0.1311
0.1343
0.34
512
0.0395
3,0383
0.0382
0.1198
0.1157
0.07
0.13
0/0376
3.0430
1024
0.1110
0.1224
0.05
-.11
2048
2.0435
0,3411
0.0415
0.033M
-004
2048
0.1261
0.1244
0.1153
0.1381
0.1328
0.10
Numbers of sub networks
Numbers
Figure 8. Comparison of the performance CSG-ODE models with different numbers and widths of subnetworks on both tasks (dynamic adjustment of learning rate).
the dynamic evolution of nodes. In order simulate the solution of differential equations via computers, numerical solvers required to approximate the solutions of these equations. The choice of numerical solver directly influences both the accuracy of the model and its computational efficiency.
Numerical solvers can generally be classified into two categories: explicit methods and implicit methods. Explicit methods are relatively simple terms of computation, as each step only depends on the current known values. As a result, they are usually faster in computation, but their accuracy may be lower. Implicit methods, on the other hand, involve not only the current known values but also require consideration of future values in each step. While this makes the computation more complex, implicit methods typically offer higher accuracy and better stability. Furthermore, solvers also be classified based their order of accuracy and their ability to adapt the step size. A higher order generally corresponds to better accuracy, while an adaptive step size allows the solver to dynamically adjust the step length based on error estimates, thereby improving computational efficiency without compromising precision.
The Euler method is a first-order explicit solver known for its simplicity and computational efficiency. Although it has lower accuracy, its simplicity makes it suitable for preliminary trials and rapid computations. Dopri5, fifth-order explicit adaptive solver variant of the Runge-Kutta method), offers higher accuracy and can adjust the step size based on estimated errors, thus providing balance between accuracy and computation time. In this experiment, used the walk capture data from Subject 35 as example to compare the impact of two solvers (Euler's method and Dopri5) the model's prediction accuracy and computational efficiency.
Table 10. Friedman test statistics
Task
ratio
Friedman statistic(x22)
Degrees of Freedom
Significance ps
40%
28.54
6
7.42x
Interpolation
60%
28.97
6
6.16x10-1
80%
28.03
6
9.28 10 -5
40%
25.80
6
2.42
Extrapolation
60%
26.40
6
1.88 1.88x10-1 10
80%
27.77
6
1.04x10-4
The results are presented in the Table The use of the Dopri5 method significantly improved the model's performance, particularly in the extrapolation task, where a substantial enhancement in performance was observed. However, compared to the Fuler method, the computational time for Dopris increased on average by 71.7%.
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CSG-ODE: ControlSynth Graph ODE for Modeling Complex Evolution of Dynamic Graphs
H.3. Statistical Significance Analyses
In order verify the robustness of the experimental results and to strengthen the empirical claims, we conducted the Friedman test on the experimental results in Table and Table 2.The statistical results of the Friedman test shown in Table 10. Among them, the significance level a, 0.05, the significance Ps much less than the significant level, which indicates the rejection of the original hypothesis (the original hypothesis is that there is no significant difference in the predictive effect of these seven models), i.e., it indicates that there is significant difference in the predictive effect of the seven models on the individual datasets. And the larger the Friedman statistic (x2), the more significant the difference in prediction results within the set.
Algorithm
We summarize the learning algorithm of our CSG-ODE in Algorithm 1.
Algorithm Training Algorithm of CSG-ODE
Input: Observation data GTos.o
Output: The parameters in the model
Initialize model parameters while not convergence do for each training sequence do
Construct the temporal graph with Equation (5)
Generate a representation of each node by Equation (7)
Generate an approximate posterior distribution potential states for each node using Equation (12) Sample the initial latent state z! of each node
Solve ODE in Equation (14) Output the trajectories using the decoder Compute the final objective, i.e., Equation (17)
Update parameters in our CSG-ODE using gradient descent
end for end while
20


Reviews:
[{"review_id": "f4JmgmyaQP", "reviewer": "Reviewer_9q63", "summary": "The paper proposes a novel model CSG-ODE and its stable variant SCSG-ODE for continuous modeling of dynamic graphs. The approach integrates a VAE framework with neural ODEs, introducing an information propagation–based inter-node importance weighting and multiple nonlinear subnetworks to capture complex node state evolution. The authors validate their claims with extensive experiments on traffic, motion capture, and simulated physical systems, demonstrating improvements on both interpolation and extrapolation tasks compared to several baselines.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your valuable comments. We have responded to each concern as follows:**\n\nA1(Response to Claims And Evidence):\n\n${L_f}(G_o^T,e{e^T})$ is the Frechet derivative with regard to $G_o^T$ and $e{e^T}$, which denotes the total transmission rate, and is used to measure the contribution of each edge in the graph to the information transfer. $||\\cdot|{|_F}$ is the Frobenius norm,$||{L_f}(G_o^T,e{e^T})|{|_F}$ denoting the total transmissibility of the graph. ${L_f}(G_o^T,e{e^T})$ is computed by finite difference approximation, where $\\beta$ denotes the step size, and the shorter the step the more accurate it is. We use the empirical equation $\\beta = \\frac{2}{N} \\times {10^{-4}}$ proposed by Noschese(2024) for $\\beta$ selection, which is based on the error analysis of the finite-difference approximation, and is able to balance the truncation and rounding errors to ensure that the total error is minimized.\n\nA2(Response to Methods And Evaluation Criteria):\n\nWe remove the well-known graph convolution Eqn (7) in our modification. We define more explicitly the mathematical symbol for Information Propagation based Inter-node Importance Weight, as described in A1.\n\nA3(Response to Theoretical Claims):\n\nA diagonal matrix $P$ is invertible if none of its diagonal elements are zero, and by the nonnegativity of the activation function we can obtain that none of the diagonal elements of the matrix $P$ is zero, and therefore the matrix $P$ is invertible. We add this in the text.\n\nA4(Response to Experimental Designs Or Analyses):\n\n$\\alpha$ is a key parameter for adjusting the effect of the sampling density mechanism, and we performed sensitivity experiments and analyzed it (see Section 4.6).\n\nA5(Response to Supplementary Material):\n\nGiven a vector-valued function $f:\\mathbb{R}^n\\to\\mathbb{R}^m$, its Jacobian matrix ${J_f}(x)$ is a $m \\times n$ matrix, with each element denoting the partial derivative of $f$ with respect to the input variable: $J_f(x)=\\begin{bmatrix}\\frac{\\partial f_1}{\\partial x_1}& \\frac{\\partial f_1}{\\partial x_2}& \\cdots &\\frac{\\partial f_1}{\\partial x_n}\\\\\\ \\frac{\\partial f_2}{\\partial x_1}&\\frac{\\partial f_2}{\\partial x_2}&\\cdots&\\frac{\\partial f_2}{\\partial x_n}\\\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\\\ \\frac{\\partial f_m}{\\partial x_1}&\\frac{\\partial f_m}{\\partial x_2}&\\cdots&\\frac{\\partial f_m}{\\partial x_n}\\end{bmatrix}$. We have added this intermediate step in the text. Also deleted the reference to GCN in Appendix C.\n\nA6(Response to Relation To Broader Scientific Literature):\n\nThe graph neural ODE converts message propagation into differential equations, which can adaptively adjust the computational depth to avoid the over-smoothing problem of too deep GNN. Based on ODE solving, it can deal with arbitrary time intervals without predefined time step limitations, which is suitable for irregularly sampled data. Therefore, the graph neural ODE is inherently equipped with the advantages of mitigating the oversmoothing problem and dealing with irregular sampling.\n\nA7(Response to Relation To Other Strengths And Weaknesses):\n\nWeaknesses(1): To verify the correctness of the theoretical derivations, we perform additional extrapolation experiments on CSG-ODE and SCSG-ODE on walk motion data. Each model was run for 5 rounds and its mean and standard deviation were calculated $(MSE \\times{10^{-2}})$. The results show that the standard deviation of SCSG-ODE is always about half of that of CSG-ODE under all sampling ratios, which indicates that it has higher stability, thus confirming the correctness of the theoretical derivation. The experimental results are shown in the following table:\n\nratio 40% 60% 80%\n\nCSG-ODE $0.1883\\pm0.0092$ $0.1676\\pm0.0109$ $0.1524\\pm0.0097$\n\nSCSG-ODE $0.2304\\pm0.0056$ $0.1978\\pm0.0050$ $0.1787\\pm0.0043$\n\nWeaknesses(2):We provide a brief description of the Jacobian derivation $J(t)$, as described in A5. The contents of Appendix C have also been deleted.\n\nA8(Response to Other Comments Or Suggestions):\n\nSuggestion(1):We have added the following to the text for the answer in A1\n\nSuggestion(2):We have added the following to the text: Theorem: For the computation of the node importance weight matrix $D\\in\\mathbb{R}^{N\\times N}$, the time complexity is $O({N^3})$.\nNote: where $\\exp_0(G_o)$ is needed to compute the matrix indices by power series expansion, which usually requires $O({N^3})$ operations due to the matrix multiplication operations involved. The Frobenius norm and the linear operations that follow produce an additional $O({N^2})$ computation, but this is covered by the dominant term, $O({N^3})$.\n\nA9(Response to Questions For Authors):\n\nQuestion(1):See A1.\n\nQuestion(2):See A3.", "claims_and_evidence": "The paper makes several well-supported claims, notably regarding the improved modeling of time-varying node relationships and nonlinear state evolution, as evidenced by extensive experiments across multiple datasets. However, the specific impact of the β parameter on the information propagation weighting mechanism is not entirely clear. While the experimental results generally support the model’s effectiveness, additional quantitative analysis or sensitivity studies on β could provide a more comprehensive understanding.", "methods_and_evaluation_criteria": "The proposed methods are well-motivated and fit the problem domain. The combination of a VAE with neural ODEs to model irregularly sampled data is appropriate, and the use of MSE as the evaluation metric is standard for these tasks. However, some parts of the method, particularly the lengthy derivations and symbol definitions in the model formulation, could be streamlined for clarity.", "theoretical_claims": "The paper offers a stability proof for the SCSG-ODE model based on the properties of antisymmetric matrices and Jacobian eigenvalue analysis. The proof is generally correct but would benefit from clearer explanations in parts—such as the relationship between the non-negativity of the activation function’s derivative and the matrix invertibility. Overall, the theoretical claims are sound but could be refined in presentation.", "experimental_designs_or_analyses": "The experimental design is comprehensive, covering multiple datasets and tasks, and includes useful ablation studies and sensitivity analyses. The comparison with existing methods is thorough. One suggestion is to provide more detailed quantitative discussion on the effect of the sampling density adjustment mechanism to help readers better appreciate its role.", "relation_to_broader_scientific_literature": "The paper positions itself well within the current literature on graph neural networks, neural ODEs, and VAE-based time series modeling. It builds on prior work such as Latent-ODE, LG-ODE, and NRI+RNN, and its contributions are clearly distinguished. Additionally, the authors could further elaborate on how their approach mitigates common issues in existing methods—such as the over-smoothing problem in graph neural ODEs and challenges in handling irregular sampling—by contrasting their solution with recent models that specifically address these challenges.", "essential_references_not_discussed": "The manuscript cites many key works; including a discussion of alternative approaches could further enrich the context. For instance, Temporal Graph Networks (TGN) by Rossi et al. (NeurIPS 2020) offer a discrete-time framework for dynamic graph representation that contrasts with the continuous-time approach adopted here. In addition, the work on Neural Controlled Differential Equations for Irregular Time Series by Kidger et al. (NeurIPS 2020) provides insights into handling irregular sampling, which is relevant to the challenges addressed in this paper.", "other_strengths_and_weaknesses": "Strengths:\n•\tThe paper presents an innovative combination of node importance weighting with nonlinear ODE modeling. The integration of information propagation to adjust node interactions directly addresses limitations in existing approaches.\n\n•\tThe experimental evaluation is comprehensive. Detailed ablation studies and comparisons across multiple datasets clearly demonstrate the contribution of each model component.\n\n•\tThe stability proof, based on the properties of antisymmetric matrices and Jacobian eigenvalue analysis, provides concrete theoretical support for the model design.\n\nWeaknesses:\n\n•\tThe link between the theoretical stability proof and the observed empirical performance is not explicitly discussed.\n\n•\tSome areas need more clarity, such as adding brief explanations for intermediate steps in the proof (e.g., the Jacobian derivation J(t)=PA), while unnecessary basic content like \"C. Detail of GNN\" should be removed.", "other_comments_or_suggestions": "I suggest that the authors simplify parts of the derivation and include more detailed explanations for specific parameters (e.g., the selection of β). Additionally, a brief discussion of the computational overhead introduced by the node importance module would be beneficial.", "questions_for_authors": "•\tWhat considerations led to the choice of the β parameter, and how does it quantitatively affect the information propagation weighting?\n\n•\tHow does the non-negativity of the activation function’s derivative relate to matrix invertibility in your stability proof?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BwOUJtFb1O", "reviewer": "Reviewer_1rWV", "summary": "This paper focuses on graph ODE model that handles dynamic relations and nodes with non-linear state evolution. The paper proposes a model called CSG-ODE that incorporates learnable latent graphs and time-varied graph snapshots. The CSG-ODE involves multiple dynamic subgraphs to capture state change of nodes. Experimental results on different datasets validate the effectiveness of CSG-ODE.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your valuable comments. We have responded to each concern as follows:**\n\nA1(Response to Claims And Evidence):\n\nSuggestion(1): Information Propagation based Inter-node Importance Weight can better capture the time-varying relationship in the following aspects:\n- Make up for the lack of local information: by measuring the contribution of each edge to the total information propagation in the graph, we make the weight not only contain the original topological information, but also portray the influence of edges on the overall information flow, thus enhancing the model's ability to utilize the global information.\n- Explanation of real-world application: in a transportation network, the efficiency of information flow will be improved if the accessibility between nodes $i$ and $j$ is enhanced (e.g., by widening the lanes). Our approach enables the model to more accurately simulate dynamic interactions in complex networks by measuring the role of edges in overall information propagation.\n- Experimental Validation: In the ablation experiment, we remove the module. The results show that the model performance decreases significantly after removing the module, which further proves the key role of the mechanism in improving the effectiveness of the model.\n\nSuggestion(2): To verify the correctness of the theoretical derivations, we perform additional extrapolation experiments on CSG-ODE and SCSG-ODE on walk motion data. Each model was run for 5 rounds and its mean and standard deviation were calculated $(MSE \\times{10^{-2}})$. The results show that the standard deviation of SCSG-ODE is always about half of that of CSG-ODE under all sampling ratios, which indicates that it has higher stability, thus confirming the correctness of the theoretical derivation. The experimental results are shown in the following table:\n\nratio 40% 60% 80%\n\nCSG-ODE $0.1883\\pm0.0092$ $0.1676\\pm0.0109$ $0.1524\\pm0.0097$\n\nSCSG-ODE $0.2304\\pm0.0056$ $0.1978\\pm0.0050$ $0.1787\\pm0.0043$\n\nA2(Response to Questions For Authors):\n\nThe $c_i^t$ denotes the node interaction information computed by the GNN, which we consider as the external control information of the node, and is updated as shown in Eqn(18). The equation contains two ODE equations, where the second ODE equation describes the continuous evolution of interactions between nodes and other nodes. Similarly to the classical discrete GNN, we perform a continuum treatment that allows $c_i^t$ to be modeled as an ODE. Specifically, in this model, the control information $c_i^t$ is computed by the GNN, which acts as a graph information aggregator that influences the dynamic evolution of the node's own state by aggregating information from neighboring nodes and transforming it into continuous control signals.", "claims_and_evidence": "The proposed CSG-ODE is claimed to be aiming at capturing time-varying relationships. However, there is no statistical analyses or cases illustrating such characteristics. Additionally, the SCSG-ODE has been proven to represent stable dynamic systems, yet no experimental metric is given to evaluate the stability of different ODEs.", "methods_and_evaluation_criteria": "The evaluation criteria follows the common practice of researches on graph ODEs.", "theoretical_claims": "I have gone through the theoretical analysis.", "experimental_designs_or_analyses": "The experiment and analyses are conducted properly.", "relation_to_broader_scientific_literature": "The proposed CSG-ODE enhances existing graph ODEs with extra considerations regarding the dynamic and implicit relationship between nodes.", "essential_references_not_discussed": "NA.", "other_strengths_and_weaknesses": "Strengths:\n\n+ The proposed framework incoporates the external control signals in dynamic systems, which is a valuable and novel idea in graph ODEs.\n+ The comprehensive experiments illustrate the effectiveness and functionalities of the proposed CSG-ODE.\n+ Theoretical proofs and analyses are provided to support the stability of the  SCSG-ODE model.", "other_comments_or_suggestions": "NA.", "questions_for_authors": "1. It seems that the control $c_i^t$ is updated via ODEs that fully contolled by $z_t$. Could you explain how the $c_i^t$ capture the external information of the dynamic systems?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wBlBbEBQ9s", "reviewer": "Reviewer_QZ2k", "summary": "The paper introduces the ControlSynth Graph (CSG-ODE) model, which improves iupon existing Graph Neural ODE models for dynamic graph representation. The model incorporates node importance weights based on information propagation and employs multiple subnetworks with nonlinear activation functions to better capture the nonlinear evolution of node states. Additionally, the paper presents an extension, Stable CSG-ODE (SCSG-ODE), which theoretically improves model stability. \n\nExtensive experimental evaluations on several dynamic system datasets demonstrate that CSG-ODE outperforms existing models, while SCSG-ODE excels in both stability and performance, particularly in high-stability scenarios.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your valuable comments. We have responded to each concern as follows:**\n\nA1(Response to Methods And Evaluation Criteria):\n\nThe mechanism improves model performance in the following ways:\n- Make up for the lack of local information: by measuring the contribution of each edge to the total information propagation in the graph, we make the weight not only contain the original topological information, but also portray the influence of edges on the overall information flow, thus enhancing the model's ability to utilize the global information.\n- Explanation of real-world application: in a transportation network, the efficiency of information flow will be improved if the accessibility between nodes $i$ and $j$ is enhanced (e.g., by widening the lanes). Our approach enables the model to more accurately simulate dynamic interactions in complex networks by measuring the role of edges in overall information propagation.\n- Experimental Validation: The results of the ablation experiments show a significant decrease in model performance after the remove of this module, further demonstrating the key role of this mechanism in improving the effectiveness of the model.\n\nA2(Response to Experimental Designs Or Analyses):\n\nWe performed Friedman's test (significance level $\\alpha_s = 0.05$):\n\nTask ratio Friedman statistic\n\nInterpolation 40% 28.54\n\n~ 60% 28.97\n\n~ 80\\% 28.03\n\nExtrapolation 40% 25.80\n\n~ 60% 26.40\n\n~ 80% 27.77\n\nThe critical value of the Friedman test is 2.508 when $k_s = 7,\\alpha_s = 0.05$, and $N_s = 5$, where $k_s$ denotes the number of models compared and $N_s$ denotes the number of data sets. Since the calculated Friedman statistic is much larger than the critical value, there is a significant difference in the predictive performance among the seven models. In addition, the larger the Friedman statistic, the more significant the difference in prediction results.\n\nA3(Response to Supplementary Material):\n\nTo verify the correctness of the theoretical derivations, we perform additional extrapolation experiments on CSG-ODE and SCSG-ODE on walk motion data. Each model was run for 5 rounds and its mean and standard deviation were calculated $(MSE \\times{10^{-2}})$. The results show that the standard deviation of SCSG-ODE is always about half of that of CSG-ODE under all sampling ratios, which indicates that it has higher stability, thus confirming the correctness of the theoretical derivation. The experimental results are shown in the following table:\n\nratio 40% 60% 80%\n\nCSG-ODE $0.1883\\pm0.0092$ $0.1676\\pm0.0109$ $0.1524\\pm0.0097$\n\nSCSG-ODE $0.2304\\pm0.0056$ $0.1978\\pm0.0050$ $0.1787\\pm0.0043$\n\nA4(Response to Relation To Other Strengths And Weaknesses):\n\nWeaknesses(1): See A3.\n\nWeaknesses(2): We performed a more detailed error analysis for each dataset. For real datasets, numerical errors in ODE solving are unavoidable. In addition, real-world data are affected by complex factors such as environmental changes that are not taken into account and may exacerbate the accumulation of errors. In the ablation experiments, we further analyze the effects of each component. Among them, the Ours-no EI performance is significantly reduced because the weights not only encode the topology of the original graph, but also reflect the role of edges in the global information propagation, which improves the model's ability to portray time-varying relationships among nodes.\n\nA5(Response to Other Comments Or Suggestions):\n\nSuggestion(1):${L_f}(G_o^T,e{e^T})$ is the Frechet derivative with regard to $G_o^T$ and $e{e^T}$, which denotes the total transmission rate, and is used to measure the contribution of each edge in the graph to the information transfer. $|| \\cdot |{|_F}$ is the Frobenius norm,$||{L_f}(G_o^T,e{e^T})|{|_F}$ denoting the total transmissibility of the graph. ${L_f}(G_o^T,e{e^T})$ is computed by finite difference approximation, where $\\beta$ denotes the step size.\n\nSuggestion(2): We use the empirical Eqn.$\\beta = \\frac{2}{N} \\times {10^{ - 4}}$ proposed by Noschese(2024) for $\\beta$ selection, which is based on an error analysis of finite-difference approximation and balances the truncation and rounding errors to ensure that the total error is minimized. For $\\alpha$, we performed a sensitivity analysis (see Section 4.6). We finally chose to conduct most of the experiments with $\\alpha=0.5$ because in the extrapolation experiments, the results around $\\alpha=0.5$ tend to be stable and locally optimal.\n\nA6(Response to Questions For Authors):\n\nIn large dynamic graphs, it may be difficult to adequately capture the complex interaction patterns between nodes by relying only on GNN as graph information aggregators. Therefore, we plan to construct modeling methods that are more in line with real information propagation in order to model dynamic interaction processes more effectively. In addition, we plan to incorporate external environmental variables to enhance the generalization ability of the model.", "claims_and_evidence": "Yes. \n\nThe claims regarding the stability improvements in SCSG-ODE are theoretically supported.\n\nThe experimental validation of these stability improvements could be more explicitly demonstrated.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are well-suited to the problem.\n\nHowever, one concern is how the node importance mechanism specifically enhances performance compared to models that rely solely on latent node representations. For instance, in dynamic graphs with complex node relationships, does incorporating node importance weighting lead to a significant performance improvement?", "theoretical_claims": "yes. \n\nThe theoretical claims regarding the stability of SCSG-ODE are supported by a theoretical proof provided in the appendix.", "experimental_designs_or_analyses": "yes.\n\nHowever, the current experimental results lack statistical significance tests, raising concerns about whether the proposed method is statistically superior to other approaches. Including statistical significance analyses (e.g., confidence intervals, hypothesis testing, or variance analysis) would help validate the robustness of the results and strengthen the empirical claims.", "relation_to_broader_scientific_literature": "By introducing adaptive node importance and nonlinear ODEs, this paper improves upon existing methods that primarily rely on linear dynamics. These enhancements enable the model to capture more complex temporal dependencies and better represent the evolving structures of dynamic graphs, addressing limitations of traditional approaches.", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "Strengths:\n\n\t1.\tThe introduction of node importance weighting and nonlinear dynamics in ODE-based graph modeling is novel and promising, offering a more expressive approach to dynamic graph representation.\n\n\t2.\tThe experimental results are convincing, demonstrating that CSG-ODE outperforms existing methods across multiple datasets, supporting the model’s effectiveness.\n\n\t3.\tThe theoretical contribution regarding stability in SCSG-ODE provides a valuable addition to the literature, offering insights into stability improvements in dynamic graph ODEs.\n\nWeaknesses:\n\n\t1.\tWhile the stability of SCSG-ODE is theoretically proven, the experimental validation of stability improvements remains insufficient, requiring more empirical evidence to support the theoretical claims.\n\n\t2.\tCertain aspects of the experimental analysis, such as error analysis and the impact of individual model components, could be more detailed to provide a deeper understanding of the model’s behavior and limitations.", "other_comments_or_suggestions": "Some of the mathematical notation could be clarified, such as the information propagation-based inter-node importance weighting mechanism.\n\nAdditionally, a clearer description of the hyperparameter tuning process, especially regarding stability-related parameters, would improve the reproducibility and interpretability of the model.", "questions_for_authors": "What are the potential limitations of the SCSG-ODE model in terms of scalability to very large dynamic graphs, and how might these limitations be addressed in future work?\n\n\n## update after rebuttal\n\nThank you for the detailed rebuttal, which addressed my main concerns. I am pleased to recommend Accept.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "n/a"}}, {"review_id": "fg58dv2YFz", "reviewer": "Reviewer_xUJZ", "summary": "The paper proposes a new approach called CSG-ODE for modeling the evolution of dynamic graphs. The main contribution lies in introducing an information transmission-based inter-node importance weighting mechanism and utilizing nonlinear activation functions in the ODE-based modeling. The authors claim that this approach improves the stability and performance of dynamic graph models, particularly in traffic, motion capture, and simulated physical systems. The paper also presents an extension of CSG-ODE, termed Stable CSG-ODE, which theoretically guarantees enhanced stability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your valuable comments. We have responded to each concern as follows:**\n\nA1(Response to Claims And Evidence):\n\nSuggestion(1): To verify the correctness of the theoretical derivations, we perform additional extrapolation experiments on CSG-ODE and SCSG-ODE on walk motion data. Each model was run for 5 rounds and its mean and standard deviation were calculated $(MSE \\times{10^{-2}})$. The results show that the standard deviation of SCSG-ODE is always about half of that of CSG-ODE under all sampling ratios, which indicates that it has higher stability, thus confirming the correctness of the theoretical derivation. The experimental results are shown in the following table:\n\nratio 40% 60% 80%\n\nCSG-ODE $0.1883\\pm0.0092$ $0.1676\\pm0.0109$ $0.1524\\pm0.0097$\n\nSCSG-ODE $0.2304\\pm0.0056$ $0.1978\\pm0.0050$ $0.1787\\pm0.0043$\n\nQuestion(2): We believe that it depends on the demand for the task, and in experiments where stability is important, a tiny fraction of the model performance can be sacrificed in favor of SCSG-ODE.\n\nA2(Response to Methods And Evaluation Criteria):\n\nSee A1.\n\nA3(Response to Theoretical Claims):\n\nDefinition of antisymmetric matrix: The matrix $A$ satisfies ${A^T} = - A$, i.e., $a_{ij}=-a_{ji}\\forall i,j$. For any matrix $A$, it can be converted to an antisymmetric matrix by $A-A^T$ (given in Remark 3.1 in the text). In our experiments, we learn a matrix $A$ and ensure its antisymmetry with the help of methods in Remark 3.1. Antisymmetric matrices have important mathematical properties and physical significance in linear algebra, and their purely imaginary eigenvalue property ensures the stability of ODE systems.\n\nA4(Response to Experimental Designs Or Analyses):\n\nQuestion(1): We model the node state evolution using a dynamical system control framework (Eqn.(18)), where ${A_0}z_i^t$ describes the linear dynamics, and $c_i^t$ (the neighborhood interaction information computed by the GNN) is transformed into control inputs (functionally analogous to the control matrix $B$ in the classical control) by the neural network $g(\\cdot)$. Different from the classical control theory, we introduce subnetworks with nonlinear activation functions to portray the nonlinear evolution of the node states, so that the node states are driven by a combination of linear evolution, nonlinear evolution and external control.\n\nQuestion(2): We performed a more detailed error analysis of the model for each dataset. For real datasets, numerical errors in ODE solving are inevitable. In addition, real-world data are often affected by complex factors, such as environmental changes, that were not included in our consideration and may further exacerbate error accumulation.\n\nA5(Response to Supplementary Material):\n\nWe have added the training algorithm:\n\n\\STATE {\\bfseries Input:}Observation data\n\n\\STATE {\\bfseries Output:}The parameters in the model\n\n\\STATE Initialize model parameters\n\n\\WHILE {not convergence} \n\n\\FOR {each training sequence}\n\n\\STATE Construct the temporal graph with Eqn. 5\n\n\\STATE Generate a representation of each node by Eqn. 8 to 11\n\n\\STATE Generate an approximate posterior distribution of potential states for each node using Eqn.16\n\n\\STATE Sample the initial latent state $z_i^0$ of each node\n\n\\STATE Solve our ODE in Eqn. 18\n\n\\STATE Output the trajectories using the decoder\n\n\\STATE Compute the final objective, Eqn. 21\n\n\\STATE Update parameters in our CSG-ODE using gradient descent\n\n\\ENDFOR\n\n\\ENDWHILE\n\nA6(Response to Relation To Broader Scientific Literature):\n\nWe add the following discussion: Yıldız et al. (2022) accurately decompose the independent dynamics of individual objects from their interactions and infer the independent dynamics and its interaction with reliable uncertainty estimates using ordinary differential equations for underlying Gaussian processes. \n\nA7(Response to Relation To Other Strengths And Weaknesses):\n\nSee A1.\n\nA8(Response to Other Comments Or Suggestions):\n\nSuggestion(1):We put the results on the SCSG-ODE into the main text, removing the description of the GNN in the appendix, while simplifying the description of the dataset and the baseline methodology.\n\nSuggestion(2): We have changed the abstract to read that for scenarios with stability requirements or prediction tasks...\n\nA9(Response to Questions For Authors):\n\nQuestion(1): See A5.\n\nQuestion(2): $\\beta$ denotes the step size of the finite difference approximation, and we use the empirical Eqn.$\\beta = \\frac{2}{N} \\times {10^{ - 4}}$ proposed by Noschese(2024) for $\\beta$ selection, which is based on an error analysis of finite-difference approximation and balances the truncation and rounding errors to ensure that the total error is minimized. For $\\alpha$, we performed a sensitivity analysis (see Section 4.6). We finally chose to conduct most of the experiments with $\\alpha=0.5$ because in the extrapolation experiments, the results around $\\alpha=0.5$ tend to be stable and locally optimal.", "claims_and_evidence": "The provided evidence is mostly convincing, with the authors demonstrating strong experimental results comparing CSG-ODE against baseline models. However, the claims about stability improvements should be further supported by analyzing the model's performance under specific conditions. In which concrete scenarios might this method exhibit relatively better stability?", "methods_and_evaluation_criteria": "The proposed method is sound and appropriately chosen for the problem of dynamic graph modeling.\nIn the experimental results presented in Appendix Table 5, the proposed methods SCSG-ODE and CSG-ODE demonstrate advantages over other baselines, but the stability advantages of SCSG-ODE are not directly evident from these results.", "theoretical_claims": "The theoretical aspects of the paper, particularly the stability guarantee provided by SCSG-ODE, are interesting and well-supported. However, some of the notation and mathematical details could benefit from additional clarity. For instance, the explanation of the learnable anti-symmetric weight matrices in SCSG-ODE is not fully accessible to readers who may not be familiar with the underlying mathematical principles. A more intuitive explanation of why this form enhances stability would help strengthen the theoretical contribution.", "experimental_designs_or_analyses": "The experimental design is sound, with a well-chosen set of datasets and baseline models for comparison. The paper does a good job of demonstrating the effectiveness of CSG-ODE and SCSG-ODE in real-world applications. However, the analysis could be more comprehensive in certain areas. For example, while the authors mention the use of a control function for modeling node interactions, they do not provide sufficient insight into how this control information interacts with other parts of the model. Additionally, a more detailed error analysis for each dataset would help clarify the strengths and weaknesses of the proposed method.", "relation_to_broader_scientific_literature": "The authors place their work in the context of ODE-based models for dynamic graph learning, which have gained traction in recent years. The contribution of CSG-ODE lies in its combination of latent space modeling and graph-based learning. The addition of the inter-node importance weighting mechanism further differentiates it from previous methods like LatentODE and GODE. Moreover, the extension to SCSG-ODE contributes to the literature by addressing issues of stability, which have been underexplored in prior works.", "essential_references_not_discussed": "The authors do not discuss some recent works on graph neural networks for temporal and dynamic systems that could provide important context for their contributions. For example, the work by Yıldız et al. (2022) on learning interacting dynamical systems with latent Gaussian process ODEs offers a related approach that could inform their model’s latent space learning.", "other_strengths_and_weaknesses": "One strength of the paper lies in its integration of node importance weighting and nonlinear activation functions to capture complex node dynamics. The theoretical foundation for the model's stability represents a significant contribution to the literature, and the experimental results are relatively compelling. However, a minor weakness is that while the authors claim their model delivers a more stable solution, the stability experiments in the appendix are not very convincing.", "other_comments_or_suggestions": "The appendix of the paper contains a lot of content, some of which is redundant. Some important experimental results in the appendix should be placed in the main body as much as possible, and some content needs to be condensed or removed. For example, the relevant introduction of the dataset and baseline methods in the appendix are quite redundant and need to be expressed concisely. Also, the introduction about GNN is actually not necessary.\n\nIn addition, the abstract mentions \"For high-stability scenarios...\", but the content of the paper refers to \"To enhance stability in high-stability scenarios\" or \"the demands of high-stability scenarios,\" so the expression in the abstract is problematic.", "questions_for_authors": "The training procedure of the model lacks detailed description, particularly regarding how hyperparameters (such as weight matrices) are tuned during training. Are there any specific guidelines or best practices for selecting these hyperparameters?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper received four effective reviews, and all of them are positive. Overall, the paper is of good quality and should be accepted.
Decision: Accept (poster)