Paper: Structural Inference with Dynamics Encoding and Partial Correlation Coefficients (ICLR-2024)

Reviewer comment (Theory):
The choice of using partial correlation as the final step in reconstructing the graph structure merits further clarification.

Author response:
Many thanks for the comment. We selected the partial correlation coefficient (PCOR) as it has shown superior performance in StructInfer [3] benchmarks, especially for one-dimensional trajectories. We believe that with effective dimension extraction and dynamic embedding, PCOR can be adapted for broader applications, including directed graphs and multi-dimensional nodes. This was a key factor in developing SIDEC.

Reviewer comment (Presentation):
A suggestion for refining the paper layout is to consider rearranging Algorithms 1 and 2 before presenting Algorithms 3 and 4 in the appendix for better logical flow.

Author response:
Thank you for this suggestion. We have revised the manuscript accordingly to enhance readability and logical flow.

Reviewer comment (Theory):
This paper mainly designs and analyzes the SIDEC framework based on information bottleneck theory and partial correlation coefficient, but it does not give clear mathematical proofs and theoretical guarantees to explain how they influence each other.

Author response:
Thank you for your insightful feedback. We acknowledge the reviewer's concern regarding the lack of detailed mathematical proofs and theoretical guarantees in our paper, specifically in the context of the SIDEC framework's reliance on information bottleneck theory and partial correlation coefficient (PCOR).

Our proposed framework, SIDEC, utilizes the variational dynamics encoder (VDE) [1], a variant of the variational autoencoder (VAE), which adheres to the variational information bottleneck (VIB) [2] principle. Specifically, the VDE in our framework seeks to optimize the encoded representation $\mathbf{z}^t_i$ to be informative about the future state $v^{t+\tau}_i$ while minimizing dependency on the current state $v^t_i$, as dictated by the information bottleneck theory [3]. This process effectively compresses the representation of input trajectories, ensuring that the dynamics of each node are captured over a defined time window $\tau$.

Upon obtaining the dynamics embeddings, we utilize PCOR to infer the graph structure. PCOR's efficacy in structural inference for one-dimensional data [15], underpins its integration into our framework. The interplay between VDE and PCOR within SIDEC is thus crucial; the former captures dynamic node embeddings, while the latter leverages these embeddings for structural inference.

We recognize that a comprehensive mathematical proof elaborating on this interplay is absent from our current manuscript. The decision to defer a detailed proof to future work stems from the complexity of the theoretical underpinnings, which require a more extensive treatment than our current scope allows. Future work will focus on providing these rigorous mathematical validations, particularly elucidating the mutual influence of VDE and PCOR within the SIDEC framework.

We appreciate this opportunity to clarify our approach and its theoretical foundations. We believe that these additional details, along with our commitment to future work, reinforce the contributions of our paper and address the concerns raised. We will integrate this refined explanation into the Appendix of our manuscript to ensure our approach's theoretical aspects are both transparent and robust.

Reviewer comment (Experiments):
Experiment Datasets are all artificially generated and may not fully reflect the complexity and diversity of the real world. Therefore, the experimental results of this article may have certain biases and limitations, and further testing and evaluation on more real data sets are required.

Author response:
We would like to emphasis that obtaining comprehensive, real-world data for structural inference poses significant challenges, both in terms of cost and time, which is mentioned in the section of limitation in the submission. Often, real-world data lack complete information on node features or reliable knowledge of underlying interaction graphs, as seen in gene regulatory networks [15] and scene graphs [4]. In other cases, such as transaction networks [5] and road maps [6], data may not cover all nodes in the system.

Despite these challenges, we agree with the necessity of testing our method, SIDEC, on real-world data. To this end, we have conducted additional experiments using three widely recognized public traffic network datasets: PEMS03, PEMS04, and PEMS07 [7]. These datasets, derived from the California Caltrans Performance Measurement System (PeMS) [8], comprise data aggregated into 5-minute intervals. The adjacency matrix of the nodes is constructed by road network distance with a thresholded Gaussian kernel [7]. The following table summarizes these datasets:

| Datasets | # Nodes | # Edges | # TimeSteps | MissingRatio |
| -------- | ------- | ------- | ----------- | ------------ |
| PEMS03   | 358     | 547     | 26208       | 0.672\%      |
| PEMS04   | 307     | 340     | 16992       | 3.182\%      |
| PEMS07   | 883     | 866     | 28224       | 0.452\%      |

We resampled the data such that constructing 49 time steps of points for each trajectory, and obtained 534, 346, and 547 trajectories, respectively. We then split all datasets with ratio 8:2:2 into training sets, validation sets and test sets, and run SIDEC on all of them. Our results, presented below, demonstrate SIDEC's effective performance despite challenges such as noise, missing data, and adjacency matrix inaccuracies. It's important to note that these datasets' adjacency matrices only connect sensors on the same road, omitting alternative connecting paths, which could impact results.

|       | PEMS03            | PEMS04            | PEMS07            |
| ----- | ----------------- | ----------------- | ----------------- |
| SIDEC | $ 72.8 \pm 0.07 $ | $ 75.4 \pm 0.04 $ | $ 70.1 \pm 0.06 $ |

As shown in the table, despite the noise and missing data, and the inaccuracy in the adjacency matrices provided by the dataset, SIDEC manages to perform well and can deal with the challenges. It is notable that the adjacency matrices in these datasets just connect the sensors if they are on the same road, but they ignore the existence of paths or alleys that may also connect the sensors. Because of this, when compared with the experimental results discussed in the original submission, the performance of SIDEC is impacted. We've added these findings to Appendix D.4 of our revised manuscript. We believe this additional real-world data testing not only addresses your concerns but also significantly strengthens the validity and applicability of our research.

Reviewer comment (Evaluation):
This method combines VDB and PCOR, but the paper does not give the impact on training time after the combination.

Author response:
We appreciate your insightful observation regarding the necessity of addressing the training time in our methodology. In response to your query, it's important to clarify that PCOR does not require training and, therefore, does not directly influence the training duration of the variational dynamics encoder (VDE). Our method, SIDEC, integrates structural inference through two main components: training of VDE and subsequent inference utilizing both VDE and PCOR. Through our experimental observations, we found that the training phase of VDE is the most time-consuming aspect, accounting for approximately 98% of the total computation time. This pattern is consistent across all our experiments. We revised the manuscript to add this description in Appendix A.7.

Reviewer comment (Experiments):
SIDEC includes PCOR, and some PROR models are also mentioned in the Structural inference with correlations and partial correlations section. Why are there no comparative experiments of these models included in the experimental section?

Author response:
Thank you for this question. The decision to exclude a direct comparative analysis of PROR models in the main experimental section was made due to the inherent limitations of these models. Specifically, correlation-based structural inference methods are typically constrained to graph reconstruction of undirected graphs with one-dimensional node features, significantly limiting their application scope.

To address this, we conducted a thorough evaluation of various correlation-based methods, as an addition to the experimental results demonstrated in Appendix D.2. We evaluate the baselines and SIDEC on LN\_NS datasets, which is characterized by one-dimensional node feature and undirected underlying interaction graph. We add these methods for evaluation:

- Person Correlation (P-cor) [9]
- Distance Correlation (D-cor) [10]
- Low-order Partial Correlation (LPCOR) [11]
- Semi-partial Correlation Coefficients (SPCOR) [12]

Additionally, we incorporated PCOR and SIDEC results into this analysis for a comprehensive comparison. Our results are presented in the following table:

| Methods | LN\_NS\_15   | LN\_NS\_30    | LN\_NS\_50    | LN\_NS\_100  |
| ------- | ------------ | ------------- | ------------- | ------------ |
| P-cor   | $68.9± 0.31$ | $65.1± 0.10$  | $66.7± 0.39$  | $68.1 ±0.44$ |
| D-cor   | $67.3± 0.23$ | $66.4± 0.37$  | $65.2± 0.20$  | $65.0± 0.26$ |
| LPCOR   | $81.9± 0.60$ | $80.6 ± 1.56$ | $78.5 ± 0.98$ | $69.5± 3.01$ |
| SPCOR   | $95.0± 0.19$ | $91.8± 3.08$  | $84.0± 0.63$  | $71.5± 2.55$ |
| PCOR    | $97.3± 0.56$ | $92.0± 5.20$  | $84.8± 1.66$  | $79.2± 4.32$ |
| SIDEC   | $97.4± 0.03$ | $96.7± 0.02$  | $97.0± 0.03$  | $97.5± 0.02$ |

The results indicate that both P-cor and D-cor show subpar performance across all datasets. This can be attributed to their limitation of considering only pairwise correlations and overlooking the influence of other potentially impactful nodes. LPCOR, while better, still falls short as it calculates coefficients conditional on a subset rather than all other variables.

Interestingly, SPCOR and PCOR demonstrate comparable performance, with PCOR slightly outperforming SPCOR. This can be attributed to SPCOR's reliance on semi-partial correlation calculations, which may limit its effectiveness in identifying connections between nodes with larger degrees. Despite the variations in graph structure across the datasets, neither SPCOR nor PCOR outperforms SIDEC.

We believe these findings reinforce our approach of integrating VDE within SIDEC, which significantly enhances the overall performance of structural inference. We have updated our manuscript to include these comprehensive results in Appendix D.2, offering a broader context and validation for our methodological choices.

Reviewer comment (Experiments):
Suggestion: maybe you can adjust the combination of loss to the weighted sum of the reconstruction loss and the autocorrelation loss to adjust the impact of time window, and monitor the change of the performance.

Author response:
We greatly appreciate your suggestion regarding the adjustment of our loss function to a weighted sum of reconstruction and autocorrelation losses to better account for the impact of the time window.

Indeed, we have experimented with modifying the loss function in this manner. However, we encountered challenges in achieving convergence with this approach, particularly for longer time windows. This unexpected behavior has piqued our interest, and we are actively investigating potential solutions to this problem. We aim to better understand and possibly exploit this phenomenon in the context of our model's performance over extended time windows.

We hope these explanations satisfactorily address the concerns and suggestions of Reviewer 5DvY, and we're grateful for the insightful feedback that will guide our ongoing research. To facilitate easy identification, sections and paragraphs that have been revised in the manuscript are highlighted in blue.

Reviewer comment (Reproducibility):
There are some missing details about key elements of the procedure. How are the dynamics modeled, and how does the adjacency matrix A influence the dynamics? How is A estimated from the embedded representations? It's hinted that this is done by the partial correlations formula, but it doesn't seem to be explicitly stated.

Author response:
We would like to thank the reviewer for the constructive comments and pointing out the missing details. Here are our answers:

- **How are the dynamics modeled?** 

  As mentioned in the second paragraph of Section 4.1 in our original submission, we use the Variational Dynamics Encoder (VDE) [1], guided by the Variational Information Bottleneck (VIB) principle [2], to model dynamics. The VDE operates on the principle:

  $ \mathbf{z}^t_i = \arg\min_{\mathbf{z}^t_i} I(\mathbf{z}^t_i;v^t_i)-\mathfrak{u}\cdot I(\mathbf{z}^t_i; v^{t+\tau}_i),$ where $\mathbf{z}^t_i$ is the latent variable for node $i$ at time $t$, and $\mathfrak{u}$ is the Lagrangian multiplier that balances the trade-off between sufficiency and minimality. $v^{t}_i$ and $v^{t+\tau}_i$ represent the feature of node $i$ at time $t$ and $t+\tau$, respectively. In this work, we denote $\tau$ as the length of time window. The latent variable $z^r_i$ is utilized to model the dynamics of node $i$ at time $t$ with a refer of time window $\tau$. This approach minimizes the mutual information between the latent variable $\mathbf{z}^t_i$ (for node $i$ at time $t$) and the feature vector $v^t_i$, considering a time window $\tau$. In short, this method optimizes the trade-off between capturing relevant information and minimizing redundancy.

- **How does the adjacency matrix $A$ influence the dynamics?** 

  According to previous works on structural inference [3, 4, 5], based on the perspective of dynamical systems [6], the adjacency matrix has a crucial influence on the node dynamics, that is, the node features evolve in a observable time period $0 \leq t \leq T$, is based on the following formulation: 

  $ V^{t+1} \leftarrow \\{V^0, V^1, ..., V^t; \mathbf{A}\\},$ where $V^t$ denotes the all node features at time $t$, and $\mathbf{A}$ represents the adjacency matrix of the graph, which represents the structure of interaction between nodes in the dynamical system. With the widely-adopted Markovian assumption on dynamical systems, we can further simplify the previous formulation into: $V^{t+1} \leftarrow \\{V^t;\mathbf{A}\\},$ where the future features only depend on the present features and the structure of the system. We may further instantiate the formulation above, and with a scope on every node: $v_i^{t+1} = v_i^t+ \Delta \cdot \sum_ {j \in \mathcal{U}_i}f\Big(||v_i, v_j ||\_{\alpha}\Big) ,$
  where $\Delta$ denotes a time interval, $\mathcal{U}\_i$ is the set of nodes connected with node $i$, and $f(\cdot)$ is the state-transition function deriving to dynamics caused by the edge from node $j$ to $i$, and $||\cdot , \cdot||\_\alpha$ denotes the $\alpha$-distance. In this formulation, $\mathcal{U}\_i$ is actually derived from the adjacency matrix $\mathbf{A}$. So in all three formulations, we can see that adjacency matrix $\mathbf{A}$ plays an important role in the evolving of the system, it determines which nodes have interaction and thus make the whole system to be dynamic. 

- **How is $A$ estimated from the embedded representations?** 

  We acknowledge the oversight in our initial manuscript. The matrix $\mathbf{P}$, mentioned in Section 4.3, serves as the reconstructed adjacency matrix $\mathbf{A}$. We have revised Section 4.3 in the manuscript to explicitly state this.

Reviewer comment (Theory):
It's not intuitive to me why the node-level dynamics can be predicted from a single node's trajectory alone, without needing to know the dynamics of other nodes (which certainly influence the trajectories of the node in question). In other words, won't the accuracy of trajectory predictions be strongly affected if the VDE is applied separately to each node trajectory, rather than jointly?

Author response:
We appreciate your insight into the node-level dynamics. As we mentioned in the answer of previous question, the node features can be formulated as: $v_i^{t+1} = v_i^t + \Delta \cdot \sum_{j \in \mathcal{U}_i}f\Big(||v_i, v_j||\_{\alpha}\Big)$. This equation suggests that the future state of node $i$ is influenced by its current state and its interactions with connected nodes. One might assume that lacking a holistic view of the system could simplify this relationship to $v_i^{t+1}\sim v_i^t$. However, this is not the case.

From an information-theoretic standpoint, a node's current features encapsulate information about all influencing elements over time. This can be represented by modifying the above equation to: $v_i^{t+1} \rightarrow v_i^t + \Delta \cdot \sum_{j \in \mathcal{U}_i}f\Big(||v_i, v_j||\_{\alpha}\Big)$. Here, the future state at $t+1$ inherently contains information about its own past state and its interactions with adjacent nodes. This allows us to infer connections based on a single node’s trajectory, sufficient for structural inference. 

We tested an alternative approach where all node features were fed simultaneously to the VDE. This method, however, led to scalability issues due to the fixed input size being dependent on the total number of nodes. Additionally, this approach did not yield an increase in accuracy. Moreover, in such a setup, distinguishing which latent variable corresponds to which node becomes challenging, as the calculations for all nodes are interwoven in the encoder and decoder stages. Consequently, we chose to focus on single-node trajectories, which proved to be more efficient and effective for our purposes.

Reviewer comment (Experiments):
The authors state that scalability is a major contribution of the method. But in the Experiments, the largest network has only 250 nodes, which makes me question whether there is sufficient evidence for scalability.

Author response:
We aim to further clarify the computational complexity associated with the structural inference methods used in our study. Unlike a linear relationship, the complexity is quadratic in relation to the number of nodes. Specifically, for directed graphs, all algorithms evaluate each possible node pair, leading to a computational complexity of $O(n^2)$, where $n$ is the total number of nodes in the graph.

In our original submission (refer to Table 1), SIDEC has demonstrated its ability to efficiently process graphs with 150, 200, and 250 nodes, which translates to 22,500, 40,000, and 62,500 directed node pairs respectively. This is in contrast to baseline methods that struggle with graphs larger than 100 nodes. To bolster our claim of SIDEC's scalability, we conducted additional experiments using the NRI simulator [7] – a tool also employed by [3, 4, 5, 8, 9] for dataset generation. These experiments involved generating spring-ball systems with 500 and 1,000 balls, producing a total of 12,000 trajectories per graph, subsequently divided into training, validation, and test sets in an 8:2:2 ratio.

In these experiments, only SIDEC was able to operate effectively without encountering out-of-memory (OOM) issues, unlike other VAE-based baseline methods. The resulting average Area Under the Receiver Operating Characteristic (AUROC) scores over ten runs were:

|       | w. 500 balls      | w. 1,000 balls    |
| ----- | ----------------- | ----------------- |
| SIDEC | $ 92.8 \pm 0.04 $ | $ 90.1 \pm 0.03 $ |

These results not only demonstrate SIDEC's capability to reconstruct the structure of large graphs but also maintain high accuracy in structural inference across varying graph sizes. We have included these additional findings in Appendix D.3 of our revised paper.

Reviewer comment (Theory):
(pg. 4) You write that $\tau$ represents the length of the time window required for the system's dynamics to be considered Markovian. What do you precisely mean by this? Why would the time window affect whether or not the system is Markovian?

Author response:
In the context of the paper, when it is mentioned that $\tau$ represents the length of the time window required for the system's dynamics to be considered Markovian, it is referring to the concept of Markovianity in continuous-time Markov processes [10]. In these processes, the future state of the system is determined solely by its current state and not by how it arrived in that state. This property is essential to simplify the modeling and analysis of complex dynamical systems.

The time window $\tau$ is crucial because it sets the scale at which the dynamics are examined. For a system to be considered Markovian, the choice of $\tau$ should be such that the memory of past states does not significantly influence the future states beyond this time window. In practical terms, this means that if you know the state of the system at time $\tau$, then this information is sufficient to predict its state at time $t+\tau$ without needing to know its earlier history.

The impact of the time window $\tau$ on the Markovian nature of the system can be understood as follows:

- If $\tau$ is too short, the system's evolution may still be influenced by its immediate past, not adhering to the Markovian property.
- If $\tau$ is appropriately chosen, it allows for the dynamics of the system to 'forget' its past states, thus adhering more closely to the Markovian assumption. This is critical for the effective application of Markov models and for simplifying the analysis of the system's dynamics. This could be the case of analyzing open quantum system which somehow expresses its non-Markovian characteristic [11].

Therefore, the correct selection of $\tau$ is vital to ensure that the system behaves in a Markovian manner, which is a fundamental assumption for the modeling techniques discussed in the paper. We added Appendix A.4 to state it.

Reviewer comment (Experiments):
What is the dimension of the embedding of node-level dynamics produced by the VDE? And what is the effect of the dimension on the performance of the algorithm?

Author response:
In our original submission (Section 4.1), we detailed that the embedding dimension for node-level dynamics produced by the VDE is one. Specifically, for a given node feature $v_i^t$ (representing node $i$ at time $t$), the corresponding embedding $\mathbf{z}^t_i$ generated by the VDE also has a dimension of one.

This dimensionality is a critical aspect of our methodology, as the use of the partial correlation coefficient (PCOR) requires the input dimension for each node at every time step to be exactly one. This requirement presents a limitation in directly applying PCOR for structural inference with nodes that have multiple features at any time step. In such cases, our method is constrained by the necessity to process each feature individually due to the single-dimensional nature of the embeddings.

Reviewer comment (Experiments):
It's not clear to me why VDE is used to produce an embedding, rather than another embedding procedure. Was there any particular reason for using this method?

Author response:
In addition to the details provided in Section 4.1 of our paper, we chose the Variational Dynamics Encoder (VDE) for the following reasons:

1. **Time-Series Adaptation**:  VDE is uniquely designed for time-series data, making it exceptionally adept at capturing dynamic temporal behaviors. This feature is crucial in domains such as biophysics, finance, and other fields dealing with complex dynamical systems.
2. **Handling Complex Dynamics**: VDE excels at processing complex, non-linear dynamics. This capability offers a significant edge over linear dimensionality reduction techniques like PCA and tICA [1], which may not fully capture the intricacies of data in systems characterized by non-linear interactions.
3. **Predictive Capability**: The time-lagged methodology of VDE enables it to forecast future states of a system based on current observations. This predictive power is invaluable for understanding and anticipating the evolution of dynamic systems, while also embedding vital dynamics information within the generated latent variables.
4. **Deep Learning Integration**: As a variant of the Variational Autoencoder (VAE), VDE leverages the latest advancements in deep learning. This includes handling high-dimensional data and deriving rich, nuanced representations that might elude more traditional methods.
5. **Autocorrelation Loss**: A unique aspect of VDE is its loss function, which combines reconstruction loss with autocorrelation loss. This feature is particularly effective in capturing long-term dynamics within time-series data, addressing the common issue of trajectory under-sampling in real-world applications.

In essence, VDE's capabilities are particularly pronounced in scenarios involving complex, nonlinear, and dynamic time-series data, which are typical in structural inference of dynamical systems.

Reviewer comment (Experiments):
What are the "Springs" and "NetSim" simulations?

Author response:
The simulations utilized for generating the trajectories in the StructInfer datasets [13] are crucial for our study. Given the current unavailability of the GitHub repository and the limited documentation on the StructInfer authors' website (maybe the authors modified them to follow double-blinded review rules), we provide a brief overview of the two key simulations based on the information available:

**Springs Simulation.** This simulation, as outlined by [7], models the motion of spring-connected particles within a 2D space. In this setup:

- Nodes represent the particles.

- Edges correspond to springs, governed by Hooke's law.

- The dynamics follow a second-order ordinary differential equation: 

  $m_i\cdot x''i(t) = \sum_{j\in\mathcal{N}_i}-k\cdot\big(x_i(t)-x_j(t)\big)$, where:

  - $m_i$denotes the particle mass, assumed to be 1.
  - $k$ is the spring constant, set at 1.
  - $\mathcal{N}_i$ represents the set of neighboring nodes connected to node $i$.

- The equation is integrated to calculate $x'_i(t)$ and $x_i(t)$ at each time step $t$, resulting in 4D node features.

**NetSims Simulation.** Based on the NetSim dataset [12], this simulation represents blood-oxygen-level-dependent (BOLD) imaging data from various human brain regions. Key aspects include:

- Nodes symbolize spatial regions of interest in the brain.

- The interaction graphs define the connections between these regions.

- Dynamics are described by a first-order ODE model: 

  $x_i'(t) = \sigma \cdot \sum_{j\in \mathcal{N}_i}x_j(t)-\sigma \cdot x_i(t) + C\cdot u_i$, where:

  - $\sigma$ manages temporal smoothing and neural lag, set to 0.1 based on [12].
  - $C$ is used to regulate external input interactions and is set to zero to minimize noise.

- The model generates 1D node features at each time step from the computed $x_i(t)$.

We've included detailed descriptions of the Springs and NetSim simulations in Appendix B of our revised paper.

Reviewer comment (Writing):
What does "level of added Gaussian noise" mean? Are you changing the mean of the noise? Please be clear.

Author response:
The StructInfer dataset [13], which we utilized in our study, includes trajectories with Gaussian noise added at various levels. As detailed in the dataset's documentation, the authors introduced this noise to the generated trajectories to simulate real-world conditions more accurately. The modified node features, represented as $\tilde{v}_i^t$, are calculated using the formula: $\tilde{v}_i^t = v_i^t + \zeta \cdot 0.02 \cdot \Delta$, where $\zeta$ is a standard Gaussian random variable ($\zeta \sim \mathcal{N}(0,1)$), $v_i^t$ denotes the original feature vector of node $i$ at time $t$, and $\Delta$ represents the designated noise level. The noise levels applied range from 1 to 5 across all the original trajectories.

This addition of Gaussian noise is an important aspect of our experimental data, and we have updated our paper to include a comprehensive description of this process in Appendix B.

We extend our gratitude to Reviewer Fg5F for the thorough, constructive, and insightful feedback. It is gratifying to know that our work has piqued your interest. We sincerely hope that our responses adequately address your concerns and contribute to enhancing the quality of our paper. To facilitate easy identification, sections and paragraphs that have been revised in the manuscript are highlighted in blue.