Road Network Representation Learning with the Third Law of Geography

Thank you for your constructive comments and suggestions. We apologize for any confusion. We are pleased to inform you that all concerns have been addressed. Below are our responses to each comment.

Responses to Weaknesses:

1. To illustrate the Third Law of Geography, consider two roads surrounded by residential buildings. Even though two roads are located very far apart, according to the third law, as they have very similar geographic configurations, they should still have similar representations. We will include these discussions in the final version of the paper to provide readers with easy understandings of the proposed method.

2. In the "Spectral Negative Sampling" section, we design a negative sample compared with the positive sample (the augmented graph from the third law of geography) for better contrastive learning. The positive sample connects nodes with very similar geographic configurations. Thus if we design a negative sample which mainly connects roads with dissimilar geographic configurations, then by discriminating the anchors (the original graph) from the negative sample, we can enhance the representations and achieve that "roads with dissimilar geographic configurations should have dissimilar representations". Then we design the specific form of the negative sample, which is inspired by the sparsest cut problem (Equation 8). In particular, it could be a complete graph with a proper graph encoder (as in Section 4.5). However, leveraging the complete graph as the negative sample is very time and space consuming (, even with subgraph sampling). Then we apply spectral graph sparsification to effectively approximate the complete graph, which results in a d-regular graph. Finally, we design a d-regular graph as the negative sample, with a properly designed graph encoder.

3. Following the reviewer's suggestion, we conducted a case study comparing the First and Third Laws of Geography. The case study is included in the attached PDF under "Author Rebuttal by Authors" and will be added to the final version of the paper.

   • We randomly selected an anchor road, computed its representation similarity with other roads (according to cosine similarity), and displayed the top 10 most similar roads. The anchor road is red, the top 10 similar roads by the First Law are blue, and those by both laws are orange.
   • With only the First Law, similar roads are much closer to the anchor. With both laws, similar roads are farther but have similar geographic configurations, as shown by comparing street view images. This demonstrates that the Third Law ensures similar representations for roads with similar configurations, regardless of distance.

4. Thanks for your suggestion. We will polish and double-check the writing.

Response to the Questions:

1. (Refer to response to weakness 1)

2. (Refer to response to weakness 2)

3. (Refer to response to weakness 3)

4. Following the suggestion of the reviewer, we discuss the correlations and differences between: (1) context and geographic configuration; and (2) context-aware spatiotemporal learning and the Third Law of Geography.

   • (1) Context is usually defined as the some contents or attributes of a spatial entity, such as the spatial context of a road segment in [1] and the context of a POI [3], where the spatial context is still some attributes of a road. "Geographic configuration" is defined as "the makeup and the structure of geographic variables over some spatial neighborhood around a point," in A-Xing Zhu's paper [2] on the Third Law of Geography. In our problem, the geographic configuration of a road segment includes both its features and the features of its neighborhood, e.g., its surrounding buildings, natural environments, regions, etc. And street view images are very good proxies to describe those. In a word, the major difference is that the term context, in the literature of spatiotemporal learning, focuses only on a spatial entity itself, while geographic configuration concerns both the attributes of the spatial entity itself and its spatial neighborhood.
   • (2) Context aware learning, inspired by word2vec [4], encourages entities within a context to have similar representations, which is more similar to the First Law of Geography, "everything is related to everything else, but near things are more related than distant things". In contrast, the Third Law of Geography states that "The more similar geographic configurations of two points (areas), the more similar the values (processes) of the target variable at these two points (areas)." It maps the similarity relationship of the geographic configurations to the similarity relationship of the target variable (i.e., road representation in our paper).

   We will include the above discussions in the final version of the paper.

[1] Robust Road Network Representation Learning: When Traffic Patterns Meet Traveling Semantics. In CIKM '21.

[2] Spatial prediction based on Third Law of Geography. Annals of GIS, 2018.

[3] How is the Third Law of Geography different? Annals of GIS, 2022.

[4] Distributed representations of words and phrases and their compositionality. In ICLR.

Dear Reviewer cFij,

Thanks very much for providing the constructive and motivating feedback! Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?

Thank you!

Thank you for your detailed comments and suggestions. We apologize for any confusion. We are pleased to inform you that all concerns have been addressed. Below are our responses to each comment. As the reviewer has many concerns, we conduct more discussions to address them.

Response to Weaknesses:

1. We analyze the meanings of the third Law and point out that incorporating it into the road network representation learning is very beneficial. This challenge is in contrast with the first Law, which emphasizes the importance of distance effect. The Third Law focuses on mapping between relationships for geographical configurations in geospatial entities (road segments) to relationships of target variables (representations). It is more challenging to learn the mapping of relationships than to learn the mapping of values. More explanations can be found in our response to "Weakness 2" below.

2. We argue the novelty of our idea and method as follows. In general, the novelty of our papers is twofold: (1) we introduce the Third Law of Geography for road network representation learning; (2) we design novel methods to learn the third law in graph neural networks (GNNs). In particular, our novel designs include geographic configuration-aware graph augmentation and spectral negative sampling. The responses to the detailed questions are listed as follows.

   • a. The term "geographic configuration" is not functionalities of regions, and the Third Law of Geography is not related to hierarchical structures in cities. In particular, the definition of "geographic configuration" in [1] is "the makeup and the structure of geographic variables over some spatial neighborhood around a point.", far beyond the functionalities of regions. The street view images (SVIs), which provide the visual features of a road, also include important descriptions of a road. For example, the surface condition of a road and the number of lanes in a road (, which is not available for the majority of OSM data,). Also, the geographic configuration includes descriptions of a point itself and its neighborhood. Street view images describe both the visual features of a road and its neighborhood, e.g., its surrounding buildings, regions etc. Thus we think the street view images are a good proxy, among various current data, to describe the geographic configuration of a road. Besides, our work is not related to hierarchical structures in a city. However, we would also like to add another paragraph in the "Related Work" section to discuss the literature that the reviewer mentions.

   • b. For multi-view / modality fusion, our design is not a direct concatenation of different data. Our designs include: (1) projecting different data into another space with learnable parameters and then concatenating them (as illustrated in Section 4.1); (2) fusing information from multiple views with mutual information maximization. The first design (joint representation) is widely adopted in multi-modality fusion [8], while the second design is inspired by [9]. Both [8] and [9] demonstrate that information from different views can be fused properly in our method.

   • c. Our main contributions, as listed at the end of the Introduction in the paper, are: (1) introduction of the Third Law of Geography to road network representation; (2) geographic configuration-aware graph augmentation and spectral negative sampling in the graph contrastive learning framework. Those are our original designs and have never appeared in previous literature. As our proposed method is based on graph contrastive learning and SGC, which are preliminaries of our method, we need to put the contents in the method for those who are not familiar with these topics. Besides, the theoretical interpretation of SGC is closely related to the implementation of the Third Law of Geography, and thus we discuss why SGC can achieve our goal in Section 4.3. We believe that bridging existing theories to real-world applications is also very important and can extend the scope of existing theories.

   • d. We have two efforts on fusing the First and the Third Law (the tuning of weights are listed in the comments):

     • The two losses have some sharing parameters (module $g_{\theta_0}$), which can learn the consensus, while those modules ($g_{\theta_1}$ and $g_{\theta_2}$) that do not have sharing parameters learn the discrepancy (potential conflicts). The final representation is the aggregation of the outputs from the three modules.
     • Train the two losses jointly. For joint training, in our preliminary experiments, we have tried some multi-objective optimization techniques to balance the two terms. Specifically, we have tried the Pareto optimum [1] in neural networks to adaptively choose proper weights to balance the two terms. However, we find that (1) with this technique, the results do not have significant improvement; (2) We find the optimal weights for the two terms vary from (0.4, 0.6) to (0.6, 0.4), very close to (0.5, 0.5). Those are the reasons why we do not tune the weights and state in the last paragraph of Section 4.6 that "Therefore, we do not introduce additional hyper-parameters to adjust their weights."
     • We also conduct some empirical results on tuning the weights of the two losses, and the results on the road function prediction task are listed as follows. We introduce another hyper-parameter $\beta$ to balance the loss of the third law ($\mathcal{L}_1$) and the first law ($\mathcal{L}_2$) : $\mathcal{L} = \beta \mathcal{L}_1 + (1 - \beta) \mathcal{L}_2$ . The results show that the learning performance is not sensitive to the weight $\beta$.

   • e. Our paper does not focus on scalability issues in road network representation learning, and we do not claim our contribution on the scalability issue neither. We provide a sub-sampling trick in our implementation, which works well within our framework.

3. The effectiveness and when the third law and the first law are expected to work have been extensively discussed in previous literature [10, 11] in geographical sciences. Our paper applies the Third Law of Geography to road network representation rather than examining the Third Law itself. We would like to recommend readers to refer [11] for more details.

4. Sorry for this confusion. The meaning of those three notations is explained here. $\boldsymbol{Z}$ is the output of some GNN. $\boldsymbol{Z}^{[0]}$, $\boldsymbol{Z}^{[1]}$ and $\boldsymbol{Z}^{[2]}$ are three outputs from different GNNs as illustrated in Fig. 1, where the number $i$ in the superscript $\cdot^{[i]}$ indicates that $\boldsymbol{Z}^{[i]}$ is the output of $i$-th GNN.

5. About the experimental design:

   • a. Thanks for your suggestion. First, most baselines have their own strategies for tackling the scalability issues. These strategies are proposed according to the peculiarities of those methods. Thus it is not realistic to equip each baseline with the sub-sampling technique. Second, other baselines, such as SRN2Vec, have unrelated model structures regarding sampling, which prevents the sub-sampling trick from being applied in their methods. Third, to explore the best performances of other approaches, we did not integrate our method with other baselines and use their recommended settings in the experiments.
   • b. Following the reviewer's suggestion, we conducted a case study comparing the First and Third Laws of Geography. The case study is included in the attached PDF under "Author Rebuttal by Authors" and will be added to the final version of the paper. (1) We randomly selected an anchor road, computed its representation similarity with other roads (according to cosine similarity), and displayed the top 10 most similar roads. The anchor road is red, the top 10 similar roads by the First Law are blue, and those by both laws are orange. (2) With only the First Law, similar roads are much closer to the anchor. With both laws, similar roads are farther but have similar geographic configurations, as shown by comparing street view images. This demonstrates that the Third Law ensures similar representations for roads with similar configurations, regardless of distance.
   • c. We have discussed this issue in "2.d."

Road Function Prediction on Singapore

Road Function Prediction on NYC

[1] Multi-Task Learning as Multi-Objective Optimization. In NeurIPS 2018.

[2] MINE: Mutual Information Neural Estimation. In ICML 2018.

[3] Representation Learning with Contrastive Predictive Coding.

[4] Deep Graph Infomax. In ICLR 2019.

[5] Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere. ICML 2020.

[6] Robust Road Network Representation Learning: When Traffic Patterns Meet Traveling Semantics. CIKM 2021.

[7] Relational Fusion Networks: Graph Convolutional Networks for Road Networks. IEEE Trans. Intell. Transport. Syst. 2021.

[8] Multimodal Machine Learning: A Survey and Taxonomy. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019.

[9] Contrastive multi-view representation learning on graphs. In ICML 2020.

[10] Spatial prediction based on Third Law of Geography. Annals of GIS, 2018.

[11] How is the Third Law of Geography different? Annals of GIS, 2022.

[12] Urban2vec: Incorporating street view imagery and pois for multi-modal urban neighborhood embedding. In AAAI 2020

[13] Knowledge-infused Contrastive Learning for Urban Imagery-based Socioeconomic Prediction. In WWW 2023

1. Sorry for the confusion. We indeed use one contrastive learning objective to incorporate the Third Law of Geography. Contrastive learning, with its mathematical support from mutual information maximization [2], requires both positive samples and negative samples to learn its loss function (Eq. (4) (5)) [3, 4]. Section 4.5 demonstrates how to generate a negative sample for graph contrastive learning. Positive samples provide information that is positive to the anchor (the original graph), while negative samples provide negative information for contrast. By contrasting with both positive samples and negative samples, models can learn good representations without supervision [5]. Following this principle, we design a positive sample where roads with very similar geographic configurations are connected, and also design a negative sample, in Section 4.5 Spectral Negative Sampling, where roads with very dissimilar geographic configurations are connected. The detailed design of the negative sample is inspired by the sparsest cut problem (in spectral graph theory) and further optimized by spectral graph sparsification. This is why the subtitle of this subsection is "Spectral Negative Sampling". As the spectral negative sampling in this paper is particularly designed to learn the third law, we only prepare it for the third law. According to the principle of contrastive learning, it is also intuitive to incorporate the inversion version of the third law by negative samples.
2. If the first law (spatial proximity) and the third law (geographic configuration) give different results on similarity, the proposed model can make some tradeoffs between the two laws, when learning road representations. Possible scenarios include spatially distant roads with similar geographic configurations and spatially proximal roads with different geographic configurations. And this is why we need to "jointly consider the third law of geography and the first law of geography" mentioned by Reviewer UWD1.
3. We sample street view images (SVIs) on each road. In practice, more than 95% of roads are associated with SVIs. For those roads without SVIs associated with them, we leverage SVIs within several meters (i.e., SVIs located in the buffer zone of a road) as proxies to represent their geographic configurations. Finally, every road has several SVIs associated on with it. Although the scenarios mentioned by the reviewer are not the critical issues tackled by the current work, if the street view data are limited or the dataset contains some samples that do not follow the two geographic laws, the proposed method might face challenges in performing robustly as other related approaches do. However, these issues can be potentially addressed by designing the new version of Garner to endow it with the capability of learning with incomplete or contaminated road network data. In the future, we will keep on exploring the effectiveness of the proposed Garner by considering the valuable suggestions raised by the reviewer. We will also include the above discussions in the final version of the paper to provide readers who are interested in the proposed Garner with more possibilities to improve it.
4. The unique advantages of street view images (SVI) are listed as follows:
   • Distribution and Coverage of POIs: A major issue of POIs in our problem is that they are unevenly distributed in cities, and a large portion of road segments do not have nearby POIs. In contrast, SVIs cover almost every road in a city, and we can sample SVIs for each road segment.
   • Commercial Bias of POIs: Another issue is that POIs are provided by Internet Web Maps for map search, and thus their contents are mainly for commercial functions (e.g., restaurants), leading to poor performance in non-commercial regions such as industrial or residential areas [12, 13]. However, there are many roads outside commercial regions, where POIs lack a lot of information. In contrast, SVIs can provide meaningful information across diverse regions.
5. Following previous work [6, 7], the "Road Traffic Inference", a standard downstream task to evaluate the effectiveness of road representations, is to predict the average speed of each road, not to predict the future traffic. For static representation learning, all of the existing studies on road network do not consider the setting of predicting future traffic variations. Developing an advanced model which can effectively capture the dynamic correlations based on this paper would be our future work.

Dear Reviewer u7JA,

Thanks very much for providing the constructive and motivating feedback! Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?

Thank you!

Thanks for your reply. Could you please kindly let us know whether we have addressed some of your concerns?

We would like to do some further clarification on the novelty. In summary, this work is the first attempt to consider the Third Law of Geography for the road network representation learning task.

1. The meaning of geographic configuration in the Third Law of Geography is far beyond functionalities of regions. The definition of "geographic configuration" in [6] is "the makeup and the structure of geographic variables over some spatial neighborhood around a point". The makeup and structure of a road contains a lot of information such as its surrounding buildings, its width, its surrounding region. The richness of this information cannot be described with a simple "functionality of region." For example, the height of the surrounding building of a road will influence the traffic on a road [9], but the height is not some functionality of a region.
2. Existing literature [1, 2, 3, 4, 5] does not mention the Third Law of Geography and the incorporation of the third law into neural networks. The literature listed by the reviewer still only considers the First Law of Geography. As illustrated in the Introduction of our paper, current literature make the geospatial entities within a context (spatial neighborhoods) to be similar. In contrast, the third law does not consider any spatial distance, and it argues that spatial entities, with similar geographic configurations, should be similar, even if they are very far apart.
   • [1] considers spatial context on condition of specific functionality. It still focuses on (conditional) spatial proximity.
   • [2] considers "ncorporate geospatial proximity as a local geographical influence and relative distance differences as a non-local geographical influence" [2], which still depends on spatial proximity and distance.
   • [3] considers context, which includes a grid at a larger scale (spatial regions) and sequences (temporal sequences). But it still considers spatial proximity.
   • [4, 5] consider hierarchical structures in cities, which are not related to our work.
   • Spatial proximity and distance, as demonstrated in the Introduction of our paper, is depicted by the First Law of Geography. In contrast, the Third Law of Geography does NOT consider any spatial proximity (including distance) between two target entities, it sololy depends on geographic configurations.
3. In our method design to incorporate the third law: geographic configuration-aware graph augmentation and spectral negative sampling, they do NOT consider any spatial proximity or spatial distance between two road segments. This is why our method is able to generate similar representations for roads with very similar geographic configurations, even though they are very far apart. A case study to illustrate this has been included in the attached PDF under "Author Rebuttal by Authors".
4. The papers that you list are not doing the same research problem as ours. The research problem in our paper, as stated in the title is road network representation learning. In particular,
   • [1] is to solve "Next Location Prediction".
   • [2] is to solve "next POI recommendation" task.
   • [3] is to learn "location embedding" to solve "Next Location Prediction" and "trajectory classification". "Locations", defined in [3] are grids (or regions) in cities. Grids are pologons. In contrastive, a road a polyline, which does not surround some area.
   • [4] is to solve "Parking Availability Prediction".
   • [5] is to solve traffic prediction on sensor network [7], not road network.
   • Our paper is to solve "road network representation learning", and the downstream tasks include: road function prediction, road traffic inference and visual road retrieval. Both our pre-training task and downstream tasks are totally different from the papers you list.
   • The word "road" does not appear in the main body of [1, 3, 5]. The word "road" nearly does not appear in [2, 4].

[1] Pre-training Context and Time Aware Location Embeddings from Spatial-Temporal Trajectories for User Next Location Prediction. In AAAI 2021.

[2] Pre-training local and non-local geographical influences with contrastive learning. Knowledge-Based Systems 2023.

[3] Pre-training Contextual Location Embeddings in Personal Trajectories via Efficient Hierarchical Location Representations. In ECML-PKDD 2023.

[4] Semi-Supervised Hierarchical Recurrent Graph Neural Network for City-Wide Parking Availability Prediction. In AAAI 2020.

[5] GPT-ST- Generative Pre-Training of Spatio-Temporal Graph Neural Networks. In NIPS 2023.

[6] Spatial prediction based on Third Law of Geography. Annals of GIS, 2018.

[7] Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting. In ICLR.

[9] Street view imagery in urban analytics and gis: A review. Landscape and Urban Planning.

Thank you for your constructive comments and suggestions. We apologize for any confusion. We are pleased to inform you that all concerns have been addressed. Below are our responses to each comment.

Response to the Weaknesses:

1. We have conducted the sensitivity test on the hyper-paramter $k$ of kNN graph, and we find our method is not sensitive to $k$.

   • For implementation, we choose the kNN graph because of its high efficiency.
   • For kNN graphs, the hyper-parameter is $k$. Therefore, we conducted comprehensive hyper-parameter sensitivity tests on various datasets and metrics, and the results are reported in Section 5.3 and Appendix C.6. The results show that the $k$ does not have a significant impact the performance of various downstream tasks. In other words, our model is not sensitive to the values of $k$. In our main experiments, we also use $k=6$ on all experiments, and this setting can produce much better results than all the baselines.

2. Thanks so much for your suggestion. Currently, we only use those datasets because of the limited data sources. Specifically, street view images are very expensive and time-consuming to collect, and other sources of data (e.g., road function, and traffic speed) are unavailable in most cities. To mitigate the potential issues caused by data availability, we have considered diverse downstream real-world tasks, i.e., road function prediction, road traffic inference, and visual road retrieval, to comprehensively test the proposed method and other baselines. In the future, we will try to collect more datasets for evaluation.

3. Sorry for the confusion, we have uploaded the code in the "Supplementary Material" under "Abstract" in this webpage.

Response to Questions:

1. We achieve "roads with dissimilar geographic configurations should have dissimilar representations." by negative sampling in the contrastive learning framework. In particular, contrastive learning requires positive samples and negative samples. Thus we design a negative graph $\bar{\mathcal{G}}^{[1]}$ whose edges mainly connect roads with dissimilar geographic configurations. By discriminating the negative sample and anchor, we can achieve "roads with dissimilar geographic configurations should have dissimilar representations."

2. Following the reviewer's suggestion, we conducted a case study comparing the First and Third Laws of Geography. The case study is included in the attached PDF under "Author Rebuttal by Authors" and will be added to the final version of the paper.

   • We randomly selected an anchor road, computed its representation similarity with other roads (according to cosine similarity), and displayed the top 10 most similar roads. The anchor road is red, the top 10 similar roads by the First Law are blue, and those by both laws are orange.
   • With only the First Law, similar roads are much closer to the anchor. With both laws, similar roads are farther but have similar geographic configurations, as shown by comparing street view images. This demonstrates that the Third Law ensures similar representations for roads with similar configurations, regardless of distance.

3. For the GPU memory, the proposed method does not require more GPU memory as the number of roads (nodes) grows. Because we use a technique called subsampling (refer to Sample in Fig. 1 and the text description from line 207). In particular, for each iteration (forward and backward propagation), we only sample a subgraph with a fixed size of nodes (4000 nodes in our setting). We observe that the GPU memory usage is less than 10GB, and thus the proposed method can run on a single RTX 3090 GPU. In the implementation of graph learning (in a single graph), if the graph is not very large, we usually load the graph into the memory [1]. If the graph is extremely large and cannot be loaded into the memory, we can first sample the index of the nodes, and then read related edges and node features from the disk.

4. Thanks very much for raising this interesting question. In current literature on road network representation, the changes in road network itself have not been considered, because its changes are relatively slow, and usually require several months or years. We also follow their settings. Actually, graph neural networks can be trained on certain graphs and tested on other similar graphs, and maintain competitive results. (For example, the PPI datasets are several graphs of proteins, and learned GNN models are tested on unseen proteins to predict the label of nodes. [2,3]) Thus, if there are only some small changes, we do not need to re-train the model, we just need to update the input data and get the output representation with the same model. If there are significant changes in the road network, we can try to re-train the model, and re-train is not very time-consuming. We just need several hours to train the model on Tesla V100 GPU, which is even slower than on a single RTX 3090 GPU.

Response to Limitations:

1. (Refer to "Response to Weaknesses", 2)
2. (Refer to "Response to Questions", 4)

[1] https://docs.dgl.ai/generated/dgl.data.DGLDataset.html#dgl.data.DGLDataset

[2] Inductive Representation Learning on Large Graphs. In NeurIPS.

[3] Simple and Deep Graph Convolutional Networks. In ICML.

Dear Reviewer ggbs,

Thanks very much for providing the constructive and motivating feedback! Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?

Thank you!

The authors thank the reviewer for providing constructive and detailed comments, which may significantly improve this paper. We are pleased to inform you that all your concerns have been successfully addressed. Please see the detailed response to each of your comments listed below.

Response to the weaknesses:

1. In A-Xing Zhu's paper [1] on the Third Law of Geography, the term "geographic configuration" is defined as "the makeup and the structure of geographic variables over some spatial neighborhood around a point." In our scenario, the target variable is the output representation of roads, and thus we should include variables and features that can effectively describe the roads. The street view images (SVIs), which provide the visual features of a road, also include important descriptions of a road. For example, the surface condition of a road and the number of lanes in a road (, which is not available for the majority of OSM data,). Also, the geographic configuration includes descriptions of a point itself and its neighborhood. Street view images describe both the visual features of a road and its neighborhood, e.g., its surrounding buildings, regions etc. Therefore, we consider SVIs a suitable proxy to describe the geographic configuration of a road. To use SVIs in our model, we need to: (1) encode the images into vectorized representations; (2) aggregate multiple street view images along each road segment. To this goal, our implementation is "the averaging pool of CLIP-image embeddings of SVI along a road."

2. Following your suggestion, we have conducted an ablation study with different similarity measures. As mentioned in the paper, popular choices for similarity measure includes: norm/distance-based similarity, cosine similarity, Gaussian kernel. However, Gaussian kernel $\exp ( -\frac{(a - b)^2}{2 \sigma^2} )$ is a monotonic function of distance, and also a bijective function of distance. Thus we only conduct experiments on norm/distance-based similarity and cosine similarity. The results are listed in the following table. We observe that, with different similarity metrics, the model achieves very similar results.

   Road Function Prediction on Singapore

   Metric	Micro-F1 (%) $\uparrow$	Macro-F1 (%) $\uparrow$	AUROC (%) $\uparrow$
   norm	81.40 $\pm$ 0.30	62.45 $\pm$ 0.64	93.27 $\pm$ 0.22
   cosine	81.30 $\pm$ 0.34	62.26 $\pm$ 0.56	92.94 $\pm$ 0.21

   Road Function Prediction on NYC

   Metric	Micro-F1 (%) $\uparrow$	Macro-F1 (%) $\uparrow$	AUROC (%) $\uparrow$
   norm	82.97 $\pm$ 0.16	47.22 $\pm$ 0.42	89.30 $\pm$ 0.21
   cosine	82.95 $\pm$ 0.18	46.98 $\pm$ 0.56	89.13 $\pm$ 0.18

   Road Traffic Inference on Singapore

   Metric	MAE $\uparrow$	RMSE $\uparrow$	MAPE $\uparrow$
   norm	2.80 $\pm$ 0.03	3.52 $\pm$ 0.04	0.579 $\pm$ 0.030
   cosine	2.82 $\pm$ 0.02	3.54 $\pm$ 0.03	0.585 $\pm$ 0.024

   Road Traffic Inference on NYC

   Metric	MAE $\uparrow$	RMSE $\uparrow$	MAPE $\uparrow$
   norm	3.30 $\pm$ 0.02	4.40 $\pm$ 0.03	0.207 $\pm$ 0.002
   cosine	3.32 $\pm$ 0.02	4.44 $\pm$ 0.03	0.208 $\pm$ 0.002

3. Sorry for the confusion. Following the conventions in graph learning, the subscript of $\boldsymbol{L}\_{\boldsymbol{S}}$ indicates that this is a graph Laplacian matrix induced by some adjacency matrix $\boldsymbol{S}$, which is similar to $\boldsymbol{D}$, the degree matrix. In particular, $\boldsymbol{D}\_{\boldsymbol{S}}$ is the degree matrix of a graph with adjacency matrix $\boldsymbol{S}$, i.e., $(\boldsymbol{D}\_{\boldsymbol{S}})\_{i, i} := \sum_{i=1}^{n} \boldsymbol{S}\_{i}$, and $\boldsymbol{L}\_{\boldsymbol{S}} := \boldsymbol{D}\_{\boldsymbol{S}} - \boldsymbol{S}$. $\boldsymbol{D}\_{\mathcal{K}}$ and $\boldsymbol{L}\_{\mathcal{K}}$ are the degree matrix and graph Laplacian matrix of the complete graph $\mathcal{K}$ respectively. Equation 7 is listed as follows:

   $$usc\_{\mathcal{G}} = \min\_{\boldsymbol{x} \in \\{0, 1\\}^n - \\{\boldsymbol{0}, \boldsymbol{1}\\}} \frac{\sum_{(i, j) \in \mathcal{E}} (\boldsymbol{x}\_i - \boldsymbol{x}\_j)^2}{\sum_{(i, j)}(\boldsymbol{x}\_i - \boldsymbol{x}\_j)^2} = \min\_{\boldsymbol{x} \in \\{0, 1\\}^n - \\{\boldsymbol{0}, \boldsymbol{1}\\}} \frac{\boldsymbol{x}^T \boldsymbol{L}\_{\mathcal{G}} \boldsymbol{x}}{ \boldsymbol{x}^{T} \boldsymbol{L}\_{\mathcal{K}} \boldsymbol{x}},$$

   where $\boldsymbol{x}$ is a n-dimensional vector, $\boldsymbol{x}\_i$ is the $i$-th element of vector $\boldsymbol{x}$. $\forall i$, $\boldsymbol{x}\_i$ is either $0$ or 1, but $\boldsymbol{x}$ cannot be a vector of all $0$s or $1$s. These explanations are represented as $\boldsymbol{x} \in \\{0, 1\\}^n - \\{\boldsymbol{0}, \boldsymbol{1}\\}$. The semantic meaning of this equation is that, the sparsest cut problem partitions nodes into two subsets and $\\{0, 1\\}$ denotes the which subset a node should belong to.

Response to questions:

1. (Please refer to response to weakness 1.)
2. (Please refer to response to weakness 2.)
3. Sorry for the confusions in the draft. Our goal in Section 4.5 is to design the negative sample $\bar{\mathcal{G}}^{[1]}$ according to Equation 8 for the mutual information estimator in Equation 4 & 5, where the positive sample is $\mathcal{G}^{[1]}$. We get some inspirations to design the negative sample from minimizing the sparsest cut problem (Equation 8), but not to strictly minimize Equation 8. In particular, we find that in the sparsest cut problem $$\min\_{\boldsymbol{Z} \in \mathbb{R}^{n \times f}} {\frac{\operatorname{tr} (\boldsymbol{Z}^{T} \boldsymbol{L}\_{\boldsymbol{S}} \boldsymbol{Z})}{\operatorname{tr} (\boldsymbol{Z}^{T} \boldsymbol{L}\_{\mathcal{K}} \boldsymbol{Z})}},$$ where the complete graph $\mathcal{K}$ could be a very good negative sample if we are going to minimize the numerator. Recall that in Section 4.3, leveraging SGC (simple graph convolution) as the graph encoder can minimize the numerator, and thus the complete graph $\mathcal{K}$ is a very good negative sample. Maximizing the denominator $\operatorname{tr} (\boldsymbol{Z}^{T} \boldsymbol{L}\_{\mathcal{K}} \boldsymbol{Z})$ is non-trivial, while minimizing the denominator is easy to achieve, through another SGC encoder. Minimizing the denominator is opposite to what we want, so we use it as the negative sample.

Response to Limitations:

Thanks for your suggestion. We would like to include the following two paragraphs in the final version of the paper.

This paper is based on the First Law of Geography and the Third Law of Geography. Though the two laws are generally true, the method in this paper may fail where the two laws are not applicable. For example, the First Law may fail in extremely large areas or with limited data [1]. Also, we assume that the street view images are good proxies to describe the geographic configurations of roads.

The potential negative societal impact includes: (1) Our method requires street view images (SVIs) along roads. However, SVIs may not be updated and thus our methods may provide outdated information. Also, SVIs cannot provide everyday changes in a city; (2) our method currently does not consider adversarial attacks from data, and thus may provide incorrect information for downstream tasks if it is attacked.

[1] Spatial prediction based on Third Law of Geography. Annals of GIS, 2018.

Dear Reviewer UWD1,

Thanks very much for providing the constructive and motivating feedback! Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?

Thank you!

The authors would like to thank the reviewer for providing constructive comments. We have made the following clarifications to address your concerns.

Response to the Weaknesses:

1. The novelty of our paper is listed as follows:

   • We introduce the Third Law of Geography, a fundamental principle in geographical sciences, which has not been explored in previous studies in road network representation learning.
   • We design novel graph contrastive learning techniques, i.e., geographic configuration-aware graph augmentation and spectral negative sampling, where spectral negative sampling is inspired by the sparsest cut problem and spectral graph sparsification in spectral graph theory.
   • According to [1], the definition of "geographic configuration" is "the makeup and the structure of geographic variables over some spatial neighborhood around a point." Thus, the term "geographic configuration" requires information from both a spatial entity and its neighborhood.

   To the best of our knowledge, all of the above have not been explored in the literature of geospatial entity representation learning. We hope this clarification makes our novelty much easier to follow.

2. The original graph $\mathcal{G}^{[0]}$ and the augmented graph $\mathcal{G}^{[1]}$ contain different information from different views. Our goal is to fuse information from both views and generate better output representations of road segments. In particular, the augmented graph $\mathcal{G}^{[1]}$ is constructed according to the Third Law of Geography. As written in our paper, "Previous studies show that information from different views can be fused properly by maximizing their mutual information (MI) [2,3], which can also improve the quality of representations [3]." Inspired by these, we maximize the MI between the original graph $\mathcal{G}^{[0]}$ and the augmented graph $\mathcal{G}^{[1]}$ so as to allow the output representation to learn from the view constructed according to the Third Law of Geography.

3. Thanks for your suggestions. We do not include POIs in the paper due to the following reasons:

   • Focus on the Third Law of Geography: Our paper focuses on the Third Law of Geography. In our paper, street view images (SVIs) can already model geographic configurations and the third law very well. Thus, SVIs are sufficient to illustrate our concepts and achieve our research objectives.
   • Distribution and Coverage of POIs: We would like to argue that POIs are currently not appropriate for our problem. A major issue is observed that POIs are unevenly distributed in cities [10], and a large portion of road segments do not have nearby POIs. In contrast, SVIs cover almost every road in a city, and we can sample SVIs for each road segment.
   • Commercial Bias of POIs: Another issue is that POIs are provided by Internet Web Maps for map search, and thus their contents are mainly for commercial functions (e.g., restaurants), leading to poor performance in non-commercial regions such as industrial or residential areas [10, 11]. However, there are many roads outside commercial regions where POIs may not provide sufficient information for subsequent analytical tasks. In contrast, SVIs can provide meaningful information across diverse regions.
   • Future Work: We acknowledge that it is very interesting to identify an appropriate way to consider POIs in Garner in future work. And we will try our best to achieve this in the future.

[1] Spatial prediction based on Third Law of Geography. Annals of GIS, 2018.

[2] Learning representations by maximizing mutual information across views. In NeurIPS.

[3] Contrastive multi-view representation learning on graphs. In ICML.

[4] Graph convolutional networks for road networks. In SIGSpatial 2019.

[5] On representation learning for road networks. ACM Trans. Intell. Syst. Technol 2021.

[6] Spatial structure-aware road network embedding via graph contrastive learning. In EDBT 2023.

[7] Robust road network representation learning: When traffic patterns meet traveling semantics. In CIKM 2021

[8] Jointly contrastive representation learning on road network and trajectory. In CIKM 2022

[9] Learning effective road network representation with hierarchical graph neural networks. In KDD 2020

[10] Urban2vec: Incorporating street view imagery and pois for multi-modal urban neighborhood embedding. In AAAI 2020

[11] Knowledge-infused Contrastive Learning for Urban Imagery-based Socioeconomic Prediction. In WWW 2023

Dear Reviewer MjEv,

Thanks very much for providing the constructive and motivating feedback! Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?

Thank you!