Road Network Representation Learning with the Third Law of  Geography

Authors: haicang  zhou
weiming  huang
yile  chen
tiantian  he
gao  cong
yew-soon  ong
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-13 15:11:11

Abstract: Road network representation learning aims to learn compressed and effective vectorized representations for road segments that are applicable to numerous tasks. In this paper, we identify the limitations of existing methods, particularly their overemphasis on the distance effect as outlined in the First Law of Geography. In response,  we propose to endow road network representation with the principles of the recent Third Law of Geography. To this end, we propose a novel graph contrastive learning framework that employs geographic configuration-aware graph augmentation and spectral negative sampling, ensuring that road segments with similar geographic configurations yield similar representations, and vice versa, aligning with the principles stated in the Third Law. The framework further fuses the Third Law with the First Law through a dual contrastive learning objective to effectively balance the implications of both laws. We evaluate our framework on two real-world datasets across three downstream tasks. The results show that the integration of the Third Law significantly improves the performance of road segment representations in downstream tasks.

Main paper: 
 introduction: 
 Road networks, which form a fundamental infrastructure within urban spaces, describe the geometries and connectivity among road segments in transportation systems. Correspondingly, road networks serve as indispensable components to support numerous smart city applications, such as traffic forecasting [8, 14] , route inference [4, 20, 42] , and travel time estimation [19] . Motivated by the advancements of graph representation learning [29, 37] , the versatile utility of road networks has spurred research into developing effective and expressive representation learning methods for road networks. The objective is to derive functional and easily integrable representations of road segments that can align with the paradigm of neural network models.Road networks are inherently regarded as graphs, which allow existing methods for road network representation learning to build upon graph representation learning techniques. In particular, apart from encoding the topological information of road networks, these methods further integrate additional spatial characteristics, such as geographical distance [2] , which are unique to road networks. In these methods, the inductive bias induced by spatial characteristics is primarily based on the First Law of Geography [25] , which states that "everything is related to everything else, but near things are more related than distant things." This principle implies that spatially close road segments tend to have similar representations. For example, skip-gram based methods [3, 39, 40] define the context window based on hops among graph neighbors or spatial distance and derive road segment representations similar to word2vec [24] . Besides, graph neural networks [11, 17, 36, 43, 52] based methods [2, 15, 44] employ message passing [5, 53] and aggregation among road segments. Both types of methods result in similar representations for connected or proximal road segments [29, 43] .While generally true and applicable, the First Law of Geography does not adequately capture the complexity of urban environments [55] , particularly in terms of long-range relationships. Consequently, this limitation compromises the effectiveness of road segment representations in existing methods. This law predominantly emphasizes the distance decay effect, neglecting the influence of semantic factors of different areas on target variables [54] . To mitigate the limitation, the Third Law of Geography [54, 55] was proposed to further consider geographic configurations, stating 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)." The term geographic configuration refers to the description of spatial neighborhood (or context) around a point (area), and the term target variable is the road representation in our context. As a result, two road segments with similar geographic configurations should have similar representations, even if they are disconnected and distant.Recognizing the potential advantages of integrating the Third Law of Geography, we initiate pioneering research that combines the principles of both the First Law and the Third Law in road network representation learning for the first time. To facilitate this integration, it is essential to leverage data sources that provide comprehensive insights into geographic configurations for road segments. Existing approaches commonly utilize data from OpenStreetMap [27] , which includes relatively basic features such as coarse-grained road attributes (location, length, type, etc.). While these data support the condition required in the First Law of Geography and are subsequently processed through specialized model designs, they are insufficient to address the application of the Third Law. To enhance the understanding of geographic configurations, we propose to utilize street view images (SVIs) [7] as an additional data source. Street view images capture the visual context of roads and their surroundings, offering a more nuanced representation of geographic configurations. However, it is still non-trivial to tackle these two principles simultaneously.First, it is important to effectively enable the integration of the Third Law of Geography within the context of road networks. This law posits that similar representations are expected to be derived for road segments with similar geographic configurations. To this end, the module should capture and reflect the similarity relationships among geographic configurations in the resulting road segment representations, ensuring that road representations faithfully preserve the similarity relationships. Second, it is critical to harmonize the implications of applying both the First and Third Law of Geography in road network representation learning. The First Law emphasizes the importance of spatial proximity, while the Third Law focuses on the significance of similarity in geographic configurations, irrespective of spatial proximity. In real-world scenarios, two distant road segments might exhibit very similar geographic configurations due to similar surrounding buildings and environments. Conversely, two directly connected and proximally close road segments might present vastly different geographic configurations -for example, one adjacent to a commercial area and the other near a park. Given these two conditions, a framework is required to synthesize road segment representations that align with both principles while mitigating potential discrepancies as highlighted.To resolve these challenges, we propose a new framework, namely Geographic Law aware road network representation learning (Garner). First, to effectively enable the integration of the Third Law, we devise a graph contrastive learning framework [22, 37] tailored for road networks. This enhances conventional contrastive learning by incorporating geographic configuration-aware graph augmentation and spectral negative sampling. Specifically, we utilize SVIs to construct a geographic configuration view for road networks, which facilitates the augmentation of edge connections between road segments sharing similar geographic configurations, even if they are geospatially distant. Then, we employ a Simple Graph Convolution (SGC) [43] encoder, which has the property of implicitly reducing the differences between the representations of connected road segments [43] , thereby mapping the similarity relationships between geographic configurations to road segment representations in the contrastive learning process. To further align with the Third Law, the proposed contrastive learning framework is equipped with a novel spectral negative sampling technique. This sampling strategy can be mathematically demonstrated to support the principle of the Third Law in a reverse way, ensuring that road segments with dissimilar geographic configurations are represented distinctly. Second, to harmonize the effects of both the First Law and Third Law, we propose a dual contrastive learning objective, which contrasts the topological structure view with the geographic configuration graph view and the spatial proximity graph view. We maintain shared parameters in the SGC encoder and jointly train the contrastive losses, thus simultaneously learning the consensus and discrepancies of these two laws in a self-supervised manner by minimizing the dual contrastive objective.Our contributions can be summarized as follows.• We identify the limitations of existing methods in road network representation learning, and propose the integration of the Third Law of Geography to overcome the shortcomings.To the best of our knowledge, this is the pioneering attempt to integrate the Third Law of Geography in this area.• We develop a novel graph contrastive learning framework to model the Third Law through geographic configuration-aware graph augmentation and spectral negative sampling. Besides, we balance the influences of two geographic laws via a dual contrastive learning objective.• We conduct extensive experiments on two real-world road network datasets (i.e., Singapore and New York City), and evaluate our model on three downstream tasks. The experiments demonstrate that the integration of the Third Law significantly enhances the performance of road network representation learning.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes]Justification: The scope and topic of this paper are made at the beginning of the Abstract and Introduction. The contributions are listed at the end of the Introduction.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
related work: 
 Road network representation learning These works aim to learn representations for road segments or intersections, for various downstream tasks [4, 8, 14, 20] . Recent studies model a road network as a graph and build their method upon graph representation techniques by including geospatial information, and can be classified into two groups. Some [3, 23, 39, 40] adopt random walks to generate paths and train a skip-gram model [29] , incorporating geospatial information based on distance [39] or spatial constraints. Others [2, 15, 44, 48] use graph neural networks [17, 36] to ensure proximal or connected roads have similar representations. Some also include traffic data (e.g., GPS trajectories of vehicles [3, 32] ) to enhance the representation. The theoretical support behind these methods is the First Law of Geography [25] . However, this principle has overemphasized spatial proximity and thus [54] proposes the Third Law of Geography, which argues that geographic configurations play critical roles in geospatial data analytics. This paper pioneers research on modeling the geographic configurations and the Third Law for road network representation learning.Unsupervised graph representation learning Graph representation learning aims to learn representations for graph components like nodes, edges, or entire graphs, where unsupervised node representation learning is most relevant to ours. Despite [12, 13] on masked auto-encoding [10] , the majority relies on contrastive learning [35] or its variants, which usually comprises several components: graph augmentation, contrastive strategy, negative sampling, and a loss function (e.g., mutual information (MI) estimator). Graph augmentation [21, 45] is to generate positive [9] or negative [37] samples, though some recent studies try to remove it [26, 49] . Contrastive strategy chooses which two components to contrast, such as node-node contrast [30, 56] or node-graph contrast [9, 21] . Negative sampling is required by the loss. The loss is usually to maximize the MI [9, 37, 50] between an entity with its positive samples and minimize the MI with its negative samples. Several recent studies have also tried to develop new objectives for graphs [26, 47] . Unlike existing studies our method further encodes geographic laws and more data sources to feed the contrastive loss.3 Preliminaries and problem definition Definition 1 (Graph). A graph is denoted as G = (V, E, X), where V and E denote the set of nodes and edges respectively. Let n = |V| denote the number of nodes and m = |E| denote the number of edges. X ∈ R n×f ′ is the feature matrix, with each row X i representing the features on node i. Let A ∈ R n×n denote the adjacency matrix of G, describing the connections in E. If node i and node j are disconnected, then A i,j = 0; otherwise A i,j ̸ = 0. By adding self-loops, we define Ã = A + I.Definition 2 (Graph Laplacian matrix). The Laplacian matrix of a graph G is defined as L := D -A, where D is the diagonal degree matrix with D i,i as the degree of node i and D i,j = 0 ∀i ̸ = j. Definition 3 (Road Network). Road networks are composed of road segments and intersections (junctions) of road segments. We use the term "road" to denote road segment in later sections for brevity. Road networks can be regarded as a graph, which is denoted as G = (V, E, X, P). Each road segment is modeled as a node, and connected roads are linked by edges. X i represents the feature vector of road i. Besides, road networks contain P, which stores the geospatial locations of nodes. Problem Definition (Road network representation learning). Given road networks G = (V, E, X, C, P), where C denotes the geographic configurations, our objective is to learn a function φ : (X, C, A, P) → Z ∈ R n×f , where the i-th row Z i of Z denotes the representation of road i.We defer the definitions and discussions of geographic laws in Appendix A.1.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: The limitations can be found in Appendix D.1. The scalability issue is discussed in Appendix C.4.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
theory assumptions and proofs: 
 Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
method: 
 In this section, we present the details of Garner, which enables the integration of the Third Law of geography for learning road representations: roads with similar geographic configurations should exhibit similar representations, and vice versa. The proposed Garner is built on the recent advances in contrastive learning and spectral graph theory. Fig. 1 illustrates the framework of Garner, which consists of the following components: (1) Data preprocessing: it produces initial road features that encode the geographic configuration. (2) Graph augmentation: it generates a graph adhering to the Third Law of Geography, where roads with similar geographic configurations are connected. (3) GNN encoder: it serves as the backbone for deriving road representations from both the original graph (the topology of the road network) and the augmented graphs. (4) Graph contrastive loss: enhanced with a spectral negative sampling strategy, it effectively integrates the Third Law (the upper part in Fig. 1 ). ( 5) Dual contrastive learning objective: it aims to harmonize the effect of the First Law with the Third Law (lower part in Fig. 1 ).Garner is inspired by recent studies on contrastive learning [1, 22] showing that by maximizing the mutual information (MI) between different views, the information from these views can be fused properly, and by contrasting different views of the graph the quality of the representation can be improved [9] . By applying a contrastive loss on the augmented and original graph, Garner can effectively learn the road representation according to both the third law and the original road network, and fuse their information properly. The cross-contrast strategy also allows Garner to fuse the Third Law and the First Law with a proxy, instead of directly contrasting the laws. This allows Z [0] to learn their consensus, while Z [1] and Z [2] can learn the discrepancies of the two laws.As mentioned in Section 1, we utilize street view images (SVIs) as proxies to represent geographic configurations for road segments. Specifically, each street view image is encoded into a vectorized representation with a pre-trained image encoder (e.g., CLIP [31] ). Then, we match these SVIs to roads according to their geospatial locations. As multiple SVIs can correspond to a single road in our datasets, we aggregate the representation of these matched SVIs with average pooling in such cases. After that, the geographic configuration (GC) for road segments can be described as matrix C ∈ R n×c . We note that a small portion of roads do not align with any SVIs. For these roads, we set their GC to be the average representation of other roads. Moreover, road segments may possess other basic attributes available from OpenStreetMap, such as the road type and length. We follow previous literature [2, 3] to encode the features into a matrix X. Finally, the GC and additional road features are projected and concatenated to form H (0) = concat([CW c , XW x ]), which serves as input to our proposed framework.We propose a graph augmentation technique according to the similarity of the geographic configuration. We first define a similarity measure for geographic configuration as sim(C i , C j ). Here, we use norm-based measure sim(C i , C j ) = 1/(1 + ∥C i -C j ∥), while cosine similarity and Gaussian kernel could also be possible choices. Then, we build an augmented similarity graph based on the similarity sim(C i , C j ). In a similarity graph, similar node pairs (i.e., node pairs with large similarity scores) will be connected by edges. Popular choices for building the similarity graph include kNN graphs and threshold graphs [38] . We empirically find that the two choices produce very close results, and we therefore choose kNN graph because of their high efficiency. In a kNN graph, each node is connected with k nodes, which have the highest similarity with it, and k is a small number, so the kNN graph is very sparse. We name this process as geographic configuration aware graph augmentation and use matrix S ∈ {0, 1} n×n to denote the adjacency matrix of the augmented similarity graph G [1] .To facilitate the integration of the Third Law, we employ Simple Graph Convolution (SGC) [43] as the backbone of our graph encoder to tackle the augmented similarity graph, described as follows:EQUATIONwhereŜ = D-1/2 S S D-1/2 S, DS is the degree matrix of S := S + I, Θ is the learnable parameter, and Z ∈ R n×f is the output representation. As discussed in [43, 56] , the simple graph convolutional operation acts as a low pass filter, making the representation Z of connected nodes similar. Specifically, according to [56] , the SGC encoder in equation 1 is designed to minimize tr(Z T L S Z), which is equivalent to the following equation:EQUATION)The details of the calculation can be found in Appendix B.1. In this equation, S i,j is non-zero if and only if node i and j are connected in the augmented graph G [1] . Therefore, by minimizing tr(Z T L S Z), the difference between Z i and Z j is reduced for every connected node pair, thus aligning with the objective of the Third Law, where the representations of nodes (i.e., road segments) with similar geographic configuration are minimized. In addition, another SGC encoder with different learnable parameters is used to encode the nodes in the original graph G [0] , which represents the topological structure of road networks. As a result, we obtain two outputs, Z [0] from the original graph G [0] , and Z [1] from the augmented graph G [1] . Given the large number of road segments in a city, we adopt a sub-sampling technique to enhance the efficiency and scalability. In particular, in each iteration in training, we maintain the same set of nodes for each graph view, and only edges connecting sampled nodes are retained. The subgraphs are then processed through the graph encoders.Previous studies show that information from different views can be fused properly by maximizing their mutual information (MI) [1] , which can also improve the quality of representations [9] . Inspired by these, we maximize the MI between the original graph G [0] and the augmented graph G [1] ,EQUATIONHere |V| denotes the number of nodes in the graph.Z [0]g ∈ R f is the representation of the whole graph by applying a graph pooling operation on Z [0] . In our implementation, we choose the mean pooling. The MI estimator MI(•, •) is a widely used one based on Jensen-Shannon divergence [37] .EQUATIONMI(Z [1] i , Z [0] g ) = E (G [1] ,G [0] ) log D(Z [1] i , Z [0] g ) + E ( Ḡ[1] ,G [0] ) log(1 -D( Z[1] i , Z [0] g )) . (5)where the discriminator D is achieved by a bilinear layer (i.e., D(a, b) = a T W b ) [22] . Ḡ[0] and Ḡ [1] are negative samples required by the mutual information estimator. Specifically, Ḡ[0] is the negative sample generated by shuffling the rows of inputs H (0) , following [22, 37] , and then Z[0] is produced through the graph encoder g θ0 . We then explain how to generate Ḡ [1] .In graph contrastive learning, negative sampling is usually implemented by graph corruption such as feature shuffling [37] and edge modification [46] . In the road network context, we extend beyond these conventional methods by introducing a novel spectral negative sampling technique to integrate the Third Law of Geography. To be specific, the Third Law not only posits that "roads with similar geographic configurations should have similar representations," but also implies that "roads with dissimilar geographic configurations should have dissimilar representations." Our proposed strategy elegantly refines the objective in Equation 2in the contrastive learning process, thereby implicitly addressing the reverse implication of the Third Law. 1To achieve this, we recall that equation 2 relates closely to the objective of the sparsest cut problem (Chapter 10 of [34] and [49] ), which seeks to find cuts that minimize the number of edges between subsets of nodes. Correspondingly, nodes are densely connected within each subset, while between different subsets, nodes are sparsely connected.EQUATIONwhere the numerator denotes the number of edges across node sets S and V -S, and the denominator is the multiplication of number of nodes in the two sets. usc G can be computed asEQUATIONwhere x i is the i-th element of vector x, and K is a complete graph, which has the same nodes as G.Following [49] , we apply a continuous relaxation in this formula and extend it to the matrix form:EQUATIONwhere L S is the graph Laplacian matrix by regarding S as the adjacency matrix, and L K is the graph Laplacian matrix of the complete graph K. Minimizing equation 8 is equivalent to minimizing the numerator and meanwhile maximizing the denominator, with the same Z. Recall that the SGC encoder (equation 1) has the effect of minimizing the numerator tr(Z T L S Z). Subsequently, we design the negative sample based on Z and K. The negative sample can achieve that by discriminating positive samples from negatives, the model has the effect of maximizing the denominator tr(Z T L K Z). This can be achieved by another SGC with the same parameter as in equation 1:EQUATIONwhere A K is the adjacency matrix of K, ÂK = D -0.5A K A K D -0.5 A K = A K /n. Z = g θ1 ( ÂK , H (0)) achieves minimizing the denominator. (See the details of the computation in Appendix B.2.) Finally, regarding K (with Z as its output) as the negative sample and discriminating positive samples from it in the MI estimator (equation 4 & 5) achieve maximizing the denominator tr(Z T L K Z) in equation 8.However, performing SGC on the complete graph K entails a time and space complexity of O(n 2 ), which is computationally infeasible for large graphs. To tackle this, we further conduct an efficient approximation for the complete graph K according to spectral graph sparsification [33] .EQUATION)where K is a d-regular graph (i.e., each node has d edges connected to it) with "all of whose non-zero Laplacian eigenvalues lie between d -2 √ d -1 and d + 2 √ d -1" and each edge weight as n/d.2 Consequently, we can optimize tr(Z T L K Z) by optimizing tr(Z T L K Z). Then we simply replace ÂK in the right hand side of equation 9 with Â K and get the outputs of the negative sample asEQUATIONwhich works well in our experiments.To summarize, we generate the negative sample Ḡ [1] as a d-regular graph K with the same node set V, with the same node feature as G [1] . Then we perform SGC on the negative sample, and finally use the output representation Z to compute mutual information in 5. The negative sampling strategy is inspired by the sparest cut and can maximize tr(Z T L K Z) = 0.5n i=1 n j=1 ∥Z i -Z j ∥ 2, where the majority node pairs (i, j) have dissimilar geographic configuration, because of the sparsity of the positive sample G [1] [34] . Therefore, we can achieve the reverse implication of the Third Law -roads with dissimilar geographic configuration have dissimilar representations.While the integration of the Third Law has been effectively achieved through our module designs, the First Law remains beneficial, especially in regions that manifest identical functionality, thus encouraging representations of nearby roads within the regions to be similar. Therefore, it is important to further incorporate the inductive bias introduced by this law into the contrastive learning process. To achieve this, we adopt another graph augmentation technique based on graph diffusion [9] , which generates another graph view G [2] with the following adjacency matrix:EQUATIONwhich can be fast approximated by [18] . Then the SGC encoder g θ2 is employed to produce the output Z [2] from G [2] . The graph diffusion process connects near roads, and the SGC encoder can pull the near roads to have similar representation due to the property of SGC (as in section 4.3). Then we use the same loss for G [0] and the augmented graph G [2] .EQUATIONBy introducing this component, our Garner is endowed with a dual contrastive objective that maximizes the mutual information between the topological structure with the geographic configuration view, as well as the spatial proximity view, in alignment with the principles of these two laws. Then we train the model (Fig. 1 ) by combining L 1 and L 2 :EQUATION)We find that the model adeptly learns to balance these considerations through parameter sharing in the SGC encoder (g θ0 ), performing well without the need for manual tuning the weights of the two loss functions. Therefore, we do not introduce additional hyper-parameters to adjust their weights. During inference, we aggregate the outputs from the three graph encoders as the road segment representations for downstream tasks: Z = (Z [0] + Z [1] + Z [2] )/3. 
experiments: 
 In this section, we evaluate the proposed method and the output road representation following previous literature [2, 3] . The road representation is evaluated on three downstream tasks. We also perform a case study to show the impact of the Third Law, and ablation studies and hyper-parameter sensitivity tests to analyze the proposed method. Datasets We use data from two cities, i.e. Singapore and New York City (NYC). The datasets include road networks from OpenStreetMap (OSM, [27] ) and street view images (SVIs) from Google Map ( [7] ). The statistics can be found in Table 1 .The road representation is evaluated on three downstream tasks: road function prediction, road traffic inference, and visual road retrieval. Road function prediction is a classification task to determine the functionality of a road. Road traffic inference is a regression task predicting the average speed of vehicles on each road. Visual road retrieval involves finding the roads where the road image should be located. While road traffic inference is widely used in previous literature [2, 3, 23] , we introduce road function prediction and visual road retrieval as two new but meaningful evaluation tasks for road representations. More details for downstream tasks can be found in Appendix C.1.Evaluation metrics For road function prediction, we use Micro-F1, Macro-F1, and AUROC (the area under the ROC curve) as the evaluation metrics. For road traffic inference, we use MAE (mean absolute error), RMSE (root mean square error), and MAPE (mean absolute percentage error) as the evaluation metrics. For visual road retrieval, we use recall@10 and MRR (mean reciprocal rank).Baselines The proposed Garner is compared with seven strong baselines, including Deepwalk [29] , MVGRL [9] , CCA-SSG [47] , GGD [51] , RFN [15] , SRN2Vec [39] and SARN [2] . Some other recent methods [3, 23, 32, 44] include other data types (e.g., GPS trajectory data of vehicles) as inputs. However, GPS trajectory data are only available in very few cities, and we did not find them for one of our datasets (NYC). Thus, we cannot run these methods for comparison in our experiments.More details of the baselines can be found in Appendix C.2.Hyper-parameter settings We use the same hyper-parameters on all the datasets. The road features, and image embeddings are projected into 256 dimensions. The k = 6 in kNN graph for geographic configuration aware graph augmentation, and d = 22 for spectral negative sampling. The hyperparameters for graph diffusion is α = 0.2, as suggested by [18] . The hidden dimension and the dimension of the representation are set as 512. Other detailed settings can be found in Appendix C.3. "oom" means out-of-memory.We compare the proposed Garner with baselines in three downstream tasks, and the experimental results can be found in Table 2 , 3 and 4.In road function prediction and road traffic inference, we report the mean results and standard deviation of 30 runs for each model. Among all the tasks, Garner performs significantly better than all the baselines. Specifically, in road function prediction, Garner outperforms the best baseline by up to 22% in Micro-F1, 171% in Macro-F1, and 25% in AUROC. In road traffic inference tasks, Garner outperforms the best baselines by up to 18.5% in MAE, 17% in RMSE, and 17.4% in MAPE. This is because the geographic configuration can provide more details about the roads. For example, the geographic configurations of roads in living apartments and business regions are very different. Thus, the functions of these two roads can be more easily discriminated according to geographic configuration aware road representation. As for road traffic inference, the geographic configuration provides more details about the conditions of roads. Thus, it is beneficial for traffic systems. On visual road retrieval, all the baselines give results similar to random guesses, while Garner gives decent results. The results show that the street view images and geographic configurations provide very different information, which is not presented in road network data. But that information can be well captured by our proposed Garner. We also find that some baselines use up GPU memory and cannot run on Tesla GPUs. We discuss the scalability issue in Appendix C.4.Case study We conduct a case study comparing the First and Third Laws of Geography. The case study will show the distinctive effect of the Third Law. We randomly select an anchor road, compute its representation similarity with other roads (according to cosine similarity), and display the top 10 most similar roads. The anchor road is in red, the top 10 similar roads by the First Law are shown in blue, and those by both laws are in orange. The results can be found in Fig. 2 . With only the First Law, similar roads are much closer to the anchor. With both laws, similar roads are further 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.Figure 2 : Case study: the top 10 most similar roads found by the First Law only and both laws.We conduct ablation studies by: (1) gradually removing the components in Garner;(2) using different similarity measure to build the augmented graph. The ablation results on road function predictions are listed in Table 5 and 6, while results on other tasks can be found in Appendix C.5. We show that the street view images, utilized to describe the geographic configurations, provide significant improvements in the quality of the road representation by comparing "Garnersns -aug" and "Garner -sns -aug -SVI". We also find that explicitly modeling the Third Law of geography provides significant improvements by comparing "Garner" and "Garner -sns -aug". Spectral negative sampling also provides obvious improvements in the Macro-F1 score and visual road retrieval. As for the choice of similarity measure, kNN similarity graphs are better than threshold similarity graphs on Road Function prediction, while slightly worse on Road Traffic Inference. Meanwhile using euclidean distance generate similar results as using cosine similarity. As building kNN graphs is much faster than threshold graphs, we choose kNN graphs as the default setting. Parameter sensitivity analysis We conduct sensitivity analysis on two hyper-parameters introduced by our method. They are the degree (k) of the augmented kNN similarity graph and the degree (d) of the negative graph. The results are visualized in Fig. 3 , and more results are listed in Appendix C. 6 . In these figures, the shadows show the standard deviation. In our experiments, we find that the results are not sensitive to the hyper-parameters.  
conclusion: 
 In this paper, we embark on pioneering research investigating the Third Law of geography for road network representation learning. To model the Third Law, we introduce street view images to capture the geographic configuration and design a new contrastive learning framework with geographic configuration aware graph augmentation and spectral negative sampling. The experiments show that the proposed method brings significant improvements in road network representation and downstream tasks. There are certainly many future directions, such as modeling more geographic laws, designing new evaluations, and using other kinds of real-world data. 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification: No such risk.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: Our usage of Google Street View Images is non-commercial and academic usage for research, which is approved by Google's Guidelines (https://about.google/ brand-resource-center/products-and-services/geo-guidelines/).Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.• For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: A.1 Discussion on the Laws of Geography Definition 4 (The First Law of Geography, or Tobler's First Law of Geography, [25] ). Everything is related to everything else, but near things are more related than distant things. Definition 5 (The Third Law of Geography, [54] ). The more similar geographic configurations of two points (areas), the more similar the values (processes) of the target variable at these two points (areas).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.As extensively discussed in the paper, this study has been substantially informed by the theories in geography and geographic information science, particularly the First Law of Geography and the Third Law of Geography. In fact, our study has also been partially informed by the Second Law of Geography, which is arguably about spatial heterogeneity. The Second Law of Geography implies that geographic variables and processes exhibit uncontrolled variance [6] . We are informed by this law from the perspective that the spatial proximity graph and the original graph only connect those close enough road segments, rather than forming complete graphs. This methodological choice acknowledges the considerable heterogeneity in the influence exerted by more distant roads.B.1 Details of calculation in section 4.2tr(Z T L S Z) = k i,j Z i,k Z j,k (L S ) i,j = k ( i Z 2 i,k (L S ) i,i + i̸ =j Z i,k Z j,k (L S ) i,j ) = 1 2 k ( i Z 2 i,k (L S ) i,i + i̸ =j 2Z i,k Z j,k (L S ) i,j + j Z 2 j,k (L S ) j,j ) = 1 2 k ( i̸ =j Z 2 i,k (S i,j + 2Z i,k Z j,k (-S i,j ) + Z 2 j,k S i,j ) = 1 2 k i̸ =j S i,j (Z i,k -Z j,k ) 2 = 1 2 i,j S i,j ∥Z i -Z j ∥ 2 .B.2 Details of calculation in section 4.5Here we show thatÂK ′ K ŜK H (0) Θ = ÂK H (0) Θ. Recall that Ŝ = D-1/2 S S D-1/2S is a symmetric matrix where each row or column is summed to 1. A K is a matrix with A i,j = 1 ∀i, j. ÂK = A K /n. Thus( ÂK Ŝ) i,j = l 1 n × Ŝl,j = 1 n l Ŝl,j = 1 n .Then we have ÂK Ŝ = ÂK . (15) For the power of ÂK , we have the following result for every i and j( ÂK ÂK ) i,j = l ( ÂK ) i,l ( ÂK ) l,j = l 1 n × 1 n = 1 n ,designs relational fusion layers that can perform message passing between both graphs. However, this also significantly increases memory usage, especially on large graphs.• SRN2Vec [39] is a the road network representation method based on random walk and skip-gram model [24] . To include geospatial information for road networks, it generates random walks according to both the graph topology and geospatial distance. To incorporate the basic features from OSM (e.g., road type), it introduces additional learning objectives such as classification for road type.• SARN [2] is a road network representation framework based on graph contrastive learning. It builds a weighted adjacency matrix according to the road network topology, distance similarity, and angular similarity. It then follows GCA [57] to produce augmented graphs by dropping edges according to the weights. It also designs a negative sampling technique according to the distance. Finally, the model is trained on an InfoNCE loss [35] .We use the Adam optimizer [16] with the learning rate as 0.001 and set the training iterations as 2500 with early stopping. The sampling size is set as 4000. For the settings of baselines, we follow their default setting but set the dimension of representation as 512, the same as our method.All the code is implemented with Python=3.11.8, PyTorch=2.1 (CUDA=11.8) [28] , DGL=2.1 [41] .All the experiments are executed on a Ubuntu Server (Ubuntu 20.04), with 8 × Nvidia Tesla V100 (32GB) GPUs, Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz (40 cores and 80 threads) and 512 GB memory. The code of baselines is generally obtained from the authors' GitHub repo. The only exception is that we use the DGL's implementation (https://github.com/dmlc/dgl/tree/ master/examples/pytorch) of Deepwalk and MVGRL for better efficiency.In our experiments, we find that some of the previous works are not scalable in our datasets. The reason could be that previously, they used much smaller datasets. Specifically, the road networks in [3, 15, 39] have less than 10,000 nodes, less than one-tenth of our datasets, and thus, they do not need to consider the scalability issue. In our method, as we perform a sub-sampling process before applying graph convolution and loss, the memory usage on GPU does not grow with the data size, and thus, the proposed Garner is scalable to large road networks.The ablation studies on Road Traffic Inference and Visual Road Retrieval are listed in Table 7 and Table 8 respectively. This paper is based on the Third 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 on extremely large areas or limited data [55] .The potential negative societal impact includes: (1) Our method requires street view images (SVIs) along roads. However, SVIs may not be up-to-date 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.As discussed in the introduction, road network representation learning provides fundamental instruments for various downstream tasks in urban computing. It can improve the traffic system in cities and enhance safety. It also provides essential references for urban planners who want to know various facets of the cities. We also admit that there could be some negative societal impacts. We are committed to ensuring our models are fair, unbiased, and respectful of individuals' privacy. We also acknowledge potential risks, such as misuse of the technology.NeurIPS Paper Checklist38th Conference on Neural Information Processing Systems (NeurIPS 2024).For notation simplicity, we omit superscripts and use Z to denote Z[1] in this section.For implementation, the adjacency matrix of K will be normalized to its degree matrix, and thus the weight n/d will not change the scale of Z. 


Decision: This paper introduces a novel framework, Garner, which leverages graph contrastive learning to integrate the Third Law of Geography into road network representation learning. The method is technically sound, and the authors provide comprehensive empirical validation, showing significant improvements across several downstream tasks. While the paper is strong in novelty and effectiveness, it would benefit from clearer explanations of the Third Law's role, additional discussion of spectral negative sampling, and improved presentation. Despite these minor issues, the paper's contributions are significant, and I recommend acceptance with revisions for clarity.

Reviews: review: paper_summary: paper_summary: This paper proposes a novel framework, termed Garner, for road network representation learning that leverages both the Third Law of Geography and the First Law of Geography. It emphasizes both spatial proximity and geographic configuration similarity.  Street view images  are used to capture similarities in road surroundings by geographic configuration-aware graph augmentation, spectral negative sampling, and a dual contrastive learning objective. ​Experiments demonstrate that integrating the Third Law significantly improves performance in downstream tasks like road function prediction, traffic inference, and visual road retrieval.
strength_weakness: strength_weakness: W1: The novelty of this paper is limited, as geographic configuration has been modeled in many studies on urban computing.

W2: The choice of the lose, ie, to maximize the MI between the original graph $G^{[0]}$ and the augmented graph $G^{[1]}$, is not well justified and is difficult for me to understand.

W3: Several key factors, such as points of interest (POIs), are not taken into consideration.
questions: questions: Q1: Why maximize the MI between the original graph $G^{[0]}$ and the augmented graph $G^{[1]}$? The two graphs have different semantics.
limitations: limitations: Adequately addressed
paper_summary: paper_summary: This paper introduces Garner, dual geographic-configuration-aware graph contrastive learning framework for road representation learning. Both the third law of geography and the first law of geography are considered by using street view images, geographic configuration aware graph augmentation,  and spectral negative sampling. The effectiveness of Garner is tested on three downstream tasks:  road function classification, road traffic prediction, and visual road retrieval.
strength_weakness: strength_weakness: 1. IMHO, the emphasis of learning the third law of geography is a bit oversold. Why can the averaging pool of CLIP-image embeddings of SVI along a road represent its geographic configuration? To me, the SVI pooled embedding can be simply treated as the visual feature of each road. 
2. In terms of Section 4.2, other than the norm-based measure as the similarity measure, can you do an ablation study on different similarity measures?
3. Some math symbols are not well-explained which makes the paper a bit hard to read. For example, what are L_S, L_K, D_S, D_K? L and D are defined, but the subscriptions need to be explained. What are \mathbf{x}_i and \mathbf{x}_j in Equation 7?
questions: questions: 1. Why can the averaging pool of CLIP-image embeddings of SVI along a road represent its geographic configuration?
2. Could you do an ablation study on different similarity measures in Section 4.2?
3. Under Equation 8, you wrote "we design the negative sample based on Z and K, ..., by discriminating positive samples from negatives". Could you explain the intuition why the discriminative between positive and negative samples can minimize Equation 8?
limitations: limitations: The author need to add a paragraph about their limitations and some potential negative societal impact of their work
paper_summary: paper_summary: This paper proposes a novel method for learning representations of road networks. It highlights the limitations of existing methods that primarily use the First Law of Geography, which emphasizes spatial proximity. The authors introduce a new framework, Garner, that incorporates the Third Law of Geography, focusing on geographic configuration similarities. They employ a graph contrastive learning approach with geographic configuration-aware graph augmentation and spectral negative sampling. The framework is evaluated on real-world datasets from Singapore and New York City, showing significant performance improvements in downstream tasks.
strength_weakness: strength_weakness: 1.	In Section 4.2, the kNN and threshold-based methods for building new connections are highly sensitive to their parameters. This sensitivity greatly impacts the generation of new connections, making it difficult to control and optimize, thereby affecting the model's consistency and robustness.
2.	The study is limited to datasets from Singapore and New York City; additional testing on more diverse datasets would enhance the findings' applicability.
3.	There is no code or dataset provided.
questions: questions: 1.	How does the proposed method perform on road networks with significantly different characteristics (e.g., rural vs. urban)? As the authors said in section 4.5: "roads with dissimilar geographic configurations should have dissimilar representations."
2.	Are there any examples or case studies demonstrating the practical implementation of Garner in different urban contexts?
3.	What are the computational requirements and scalability aspects of the proposed method for larger datasets? For example, if we want to measure the similarity between different cities, we need to use national or global maps.
4.	How does the model deal with dynamically changing road networks, such as those undergoing construction or frequent changes? Does the whole model need to be re-trained after the changes?
limitations: limitations: 1.	The authors should consider testing their method on a more diverse set of datasets to ensure broader applicability.
2.	Consideration of temporal changes in road networks and how the model adapts to these changes would strengthen the practical application of the method.
paper_summary: paper_summary: This paper introduces a novel framework for road network representation learning, with the key innovation being the incorporation of the Third Law of Geography. This concept is implemented through a tailored graph contrastive learning objective, featuring geographic configuration-aware graph augmentation and spectral negative sampling. To further preserve geographic proximity (the First Law of Geography), which has been proven crucial in existing literature, the authors further propose a dual contrastive learning strategy to accommodate both modeling perspectives. Experimental results on datasets from Singapore and New York City demonstrate significant improvements over several baseline methods in road function prediction, road traffic inference, and visual road retrieval tasks.
strength_weakness: strength_weakness: 1. The research challenges identified by the paper lack depth. In particular, the first challenge is insufficiently articulated, equivalent to stating that "generating road representations that preserve the Third Law of Geography is challenging".

2. A significant weakness of the paper lies in the limited novelty of its ideas and methodological design:

   a. While existing literature may not explicitly introduce the concept of the Third Law of Geography, several works have already considered the similarity of road segments in regions with similar functionalities [1,2,3]. Moreover, in the broader field of spatio-temporal data mining, several studies have considered the hierarchical structure of urban road networks [4,5].

   b. The simple concatenation of street view image representations with other road features lacks sophistication. Images and other road attributes exist in different modalities, potentially introducing multi-modal data fusion issues that may impede the effective encoding of street view images into road representations.

   c. The Simple Graph Convolution (SGC), its theoretical interpretation, and graph contrastive learning have already been well-developed in previous research.

   d. The fusion of the third law and the first law is presented as one of the two major challenges the paper aims to address. However, the proposed solution merely combines two contrastive learning tasks without a thorough discussion of potential conflicts between them.

   e. Although the paper considers scalability issues, the proposed sub-sampling solution is relatively straightforward.

3. The paper lacks detailed discussions on the universality of the third law and the first law, particularly in large-scale road networks, raising concerns about the practical applicability of the method. Besides, merely mentioning this in the limitations section is insufficient.

4. The second paragraph at the beginning of section 4 contains confusing notations that are not previously introduced.

5. Experimental design:

   a. As the paper's solution to improve scalability is straightforward, the baseline methods should also be equipped with the same graph sampling techniques to ensure a fair comparison.

   b. The paper lacks case studies that demonstrate the effectiveness of integrating street view images. Relying solely on ablation studies is inadequate for this purpose.

   c. The paper does not use weighted average of the two contrastive learning tasks, which is important as the first law and the third law might conflict in certain scenarios. Along this line, experiments on the trade-off between the two contrastive learning tasks should also be conducted. 

[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.
questions: questions: 1. Why is the spectral negative sampling strategy only applied to the contrastive learning task corresponding to the third law? How does the derivation of this strategy in section 4.5 relate to the mutual information maximization in equation (5)? Would it not be more intuitive to design an additional contrastive learning task to focus on encouraging the node representations of the augmented graph to satisfy both the third law and its inverse version?

2. Why does the inverse version of the third law in section 4.5 hold? For instance, two roads with dissimilar geographic configurations might still exhibit spatial proximity, suggesting that their representations should remain similar.

3. What proportion of roads in the dataset are associated with street view images? How does the model perform in geographic regions where (a) street view images are limited or of poor quality, or (b) the two geographic laws do not hold?

4. Compared to widely used Points of Interest (POI) data, what are the unique advantages of street view images in this context?

5. What are the specific experimental settings for the road traffic prediction task? If the objective is to predict future traffic variations, how do static road network representations contribute to this task without considering the dynamic correlations among road segments?
limitations: limitations: see weakness
paper_summary: paper_summary: This paper proposes a novel framework for learning road network representations. Different from previous approaches, the proposed framework leverages a particularly designed graph contrastive learning method to integrate the Third Law of Geography into the process of road network representation learning, which can significantly alleviate the current shortcomings. In both classical and newly identified downstream tasks of road representation learning, the proposed framework achieves significant improvements compared with other state-of-the-art approaches.
strength_weakness: strength_weakness: W1. This paper lacks detailed explanations of using the Third Law of Geography in road network representation learning. The authors are suggested to give more explanations or examples to show the significance of using the Third Law.

W2. The spectral negative sampling is effective but may not be easy to understand for non-experts. The authors are encouraged to provide more explanations of this technique.

W3. In experiments, this paper only presents improvements in metrics, without more intuitive results. The authors are encouraged to give more discussions or empirical analysis of the implication of the third law on representations.

W4. The presentation can be improved. Also, there are some typos in the paper.
questions: questions: As stated in “Weaknesses,” I have the following questions.

Q1. Are there more detailed explanations or examples that can show the significance of the Third Law of Geography in road network representation learning?

Q2. Are there more easy-to-understand descriptions/explanations of the proposed spectral negative sampling method?

Q3. Are there case studies or discussions in the experiments that can show the implication of the Third Law of Geography in road network representation learning?

Q4. Since this work is based on the Third Law of Geography, it seems it is similar to context-aware spatiotemporal learning. It is encouraged for authors to discuss the correlations and distinguishment between context-based ST learning and Third Law of Geography based learning.
limitations: limitations: The authors have discussed the potential limitations but not much sufficient.

Comments: title: Thanks for rebuttal
Further Clarification of Novelty - 2
Further Clarification of Novelty - 1
Thanks for your reply
Thanks for your reply
Acknowledgement
Thanks for your reply
I will keep my score
Additional Questions
Additional Questions
Additional Questions
Additional Questions
Additional Questions
[Rebuttal by Authors - 2]
[Rebuttal by Authors - 5] References of Author Rebuttal and Response to Reviewer Comments
[Rebuttal by Authors - 4] Response to Questions
[Rebuttal by Authors - 3] Response to Weaknesses - Continued
[Rebuttal by Authors - 2] Results of Tuning the Weights of Losses
comment: I would like to thank the authors for their detailed responses, and my concerns are well addressed.
For the scalability and efficiency issue, we have two designs: subgraph sampling and spectral graph sparsification. The spectral graph sparsification for our spectral negative sampling is our special design and cannot be applied to other methods.
1. The subgraph sampling is to sample a much small graph from the huge road network.
2. Even with subgraph sampling, the negative sampling to encourage "the reverse implication of the Third Law: roads with dissimilar configurations should have dissimilar representations" is still very time and space consuming. It requires to sample a complete graph, with time and space complexity of $O(|V^{\prime}|^2)$, where $|V^{\prime}|$ is the number of nodes in the sampled subgraph. To improve the efficiency and scalability, we follow the theoretical results of spectral graph sparsification [8], to approximate the complete graph with a $k$-regular graph, which has much less edges, with complexity of $O(|k V^{\prime}|)$, where $d$ is the degree of nodes, and we set $k=6$ in our experiments. It is non-trival to a adopt theoretical result to negative sampling.
3. The results of spectral sparsification in our paper, is only application to approximate a complete graph. Thus it cannot be applied to other baselines, which either do not have negative sampling or do not use complete graph in negative sampling.


Thank you again to providing constructive feedbacks? Can you please let us know whether we have addressed all your questions and whether you have any additional feedback?


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[8] Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput.
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].


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[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.
Dear Reviewer MjEv,

Thanks for your reply. We also agree that careful designs are required to keep both shared information and exclusive information. And we have achieved this in our method. We would like to provide both clarifications and experimental results as follows.

1. As mentioned in [1], models can learn both shared information and exclusive information via mutual information by considering both global and local mutual information.
2. The loss in our method is one of the most widely used MI estimators,  Jensen-Shannon MI estimator [2, 3], which considers the global-local mutual information by maximizing the mutual information between a patch of an image (local high-level representations) and the image itself (high-level "global" representation).
3. In the field of **graph contrastive learning**, generating an augmented graph and using mutual information maximization to **fuse** different views of graphs, keep both shared information and exclusive information, and enhance the representation are widely adopted [4, 5, 6]. Our designs are also based on this literature to fuse different views. Besides, the experimental results in 4-6 also indicate the effectiveness of the fusion and keeping both shared information and exclusive information (i.e., geographic configurations).
4. In our **ablation study**, line 2 and line 3 in Table 5 show that the maximizing the mutual information between the original graph $\mathcal{G}^{[0]}$ and the augmented graph $\mathcal{G}^{[1]}$ to fuse different views and keep exclusive information (i.e., geographic configurations) can significantly improve the learning performance. In particular, we archive up to 25% improvement.
5. We have conducted a downstream task named **visual road retrieval** (Table 4 in the paper), aiming at finding which road a street view image should belong to. In this experiment, our method, which uses SVIs as part of the input and models the Third Law of Geography, performs significantly better than the baselines, which do not consider geographic configurations. The experimental results indicate that the representation learned a lot from the geographic configurations (exclusive information).
6. We have conducted a **case study**, which is included in the attached PDF under "Author Rebuttal by Authors". In the case study, we randomly selected an anchor road, computed its representation similarity with other roads and displayed the top 10 most similar roads. By considering the third law in our method, similar roads are farther but have similar geographic configurations. This also reveals that our method can keep exclusive information of geographic configurations.


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[1] Learning disentangled representations via mutual information estimation. In ECCV.

[2] Learning deep representations by mutual information estimation and maximization. In ICLR.

[3] Deep Graph Infomax. In ICLR.

[4] Contrastive Multi-View Representation Learning on Graphs. In ICML 2020.

[5] Graph Contrastive Learning with Adaptive Augmentation. In WWW 2021.

[6] Multi-Scale Contrastive Siamese Networks for Self-Supervised Graph
Representation Learning. In IJCAI.
I raised my score to  4: Borderline reject since my concerns are partially addressed.
Your reply does not address my concerns. A suitable loss function is the core of a learning model.

In some studies[1], models learn not only share information but also exclusive information via mutual information.

[1] Sanchez, Eduardo Hugo, Mathieu Serrurier, and Mathias Ortner. "Learning disentangled representations via mutual information estimation." Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXII 16. Springer International Publishing, 2020.
Thanks for your response. I keep my original score.
Dear Reviewer MjEv,

Thank you for your thoughtful feedback and for allowing us the opportunity to address your concerns. We are pleased to have resolved some of the issues you raised and appreciate your constructive comments.

We agree with your opinion that maximizing mutual information (MI) is appropriate for fusing views that share information. We would like to provide further clarification on the fusion of the two graphs in our approach:

1. **Shared Information Between Graphs**: Both graphs, $\mathcal{G}^{[0]}$ and $\mathcal{G}^{[1]}$, have shared information of geographic configurations as the same node features. As described in Section 4.1, we prepare their node features as concatenation of projected road features and projected street view image embeddings. Specifically, the node features in both graphs are $\boldsymbol{H}^{[0]} = \text{concat}([\boldsymbol{C} \boldsymbol{W}_c, \boldsymbol{X} \boldsymbol{W}_x])$, where $\boldsymbol{C}$ is the matrix of geographic configurations, $\boldsymbol{X}$ is the matrix of road features (from map data) and $\boldsymbol{W}_c$ and $\boldsymbol{W}_x$ are learnable projection matrix. This "joint representation" is a widely used method in multi-modality and multi-view fusion [1].
2. **Differences between Graphs**: The only difference between the original graph $\mathcal{G}^{[0]}$ and the augmented graph $\mathcal{G}^{[1]}$ are their edges, which provide different views of underlying road segments. Specifically, edges in $\mathcal{G}^{[0]}$ describe the connectivity of roads in the map, while edges in $\mathcal{G}^{[1]}$ connect roads with very similar geographic configurations. By maximizing the mutual information between them, the model can learn some "high-level factors whose influence spans multiple views" [2], for example, the similarity of road segments considering the third law. Our ablation study also demonstrates that, building the augmented graph $\mathcal{G}^{[1]}$ for contrastive learning significantly enhances the learning performance of our proposed method.

Thank you again for your valuable feedback.


[1] Multimodal Machine Learning: A Survey and Taxonomy. TPAMI.

[2] Learning Representations by Maximizing Mutual Information Across Views. In NeurIPS 2019.
I thank the authors for their detailed explanation. However, the responses on the novelty of the method and the argument on scalability are not very convincing. I would like to keep my score.
I agree with you on the use of POIs, street view images are sufficient for the illustration of your idea.
However, it's not entirely reasonable for me to say that maximizing MI aims at fusion. Maximizing MI retains only the shared information between the two graphs. In other words, the geographic information you expected may be lost during this.
I thank the authors for their further explanation and I would like to keep my score.
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?

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Dear Reviewer u7JA,

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Dear Reviewer ggbs,

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Dear Reviewer UWD1,

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Dear Reviewer MjEv,

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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.
[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.
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**
    
| $\beta$ | Micro-F1 (%) $\uparrow$ | Macro-F1 (%) $\uparrow$ | AUROC (%) $\uparrow$ |
| ------- | ----------------------- | ----------------------- | -------------------- |
| 0.1     | 81.23 $\pm$ 0.36        | 62.03 $\pm$ 0.84        | 92.90 $\pm$ 0.28     |
| 0.2     | 81.48 $\pm$ 0.37        | 62.61 $\pm$ 0.85        | 93.01 $\pm$ 0.24     |
| 0.3     | 81.22 $\pm$ 0.34        | 62.11 $\pm$ 0.62        | 92.91 $\pm$ 0.25     |
| 0.4     | 80.99 $\pm$ 0.30        | 61.50 $\pm$ 0.70        | 92.66 $\pm$ 0.26     |
| 0.5     | 81.40 $\pm$ 0.30        | 62.45 $\pm$ 0.64        | 93.27 $\pm$ 0.22     |
| 0.6     | 81.02 $\pm$ 0.44        | 60.94 $\pm$ 0.85        | 92.69 $\pm$ 0.21     |
| 0.7     | 81.19 $\pm$ 0.37        | 62.26 $\pm$ 0.88        | 92.82 $\pm$ 0.21     |
| 0.8     | 81.26 $\pm$ 0.33        | 61.91 $\pm$ 0.83        | 92.81 $\pm$ 0.23     |
| 0.9     | 81.40 $\pm$ 0.40        | 62.54 $\pm$ 0.72        | 93.05 $\pm$ 0.24     |

**Road Function Prediction on NYC**

| $\beta$ | Micro-F1 (%) $\uparrow$ | Macro-F1 (%) $\uparrow$ | AUROC (%) $\uparrow$ |
| ------- | ----------------------- | ----------------------- | -------------------- |
| 0.1     | 82.83 $\pm$ 0.21        | 46.70 $\pm$ 0.45        | 89.18 $\pm$ 0.21     |
| 0.2     | 82.93 $\pm$ 0.19        | 47.17 $\pm$ 0.37        | 89.24 $\pm$ 0.17     |
| 0.3     | 82.91 $\pm$ 0.20        | 47.25 $\pm$ 0.57        | 89.17 $\pm$ 0.20     |
| 0.4     | 82.86 $\pm$ 0.20        | 47.08 $\pm$ 0.49        | 89.14 $\pm$ 0.19     |
| 0.5     | 82.97 $\pm$ 0.16        | 47.22 $\pm$ 0.42        | 89.30 $\pm$ 0.21     |
| 0.6     | 82.97 $\pm$ 0.23        | 46.80 $\pm$ 0.56        | 89.22 $\pm$ 0.15     |
| 0.7     | 83.04 $\pm$ 0.15        | 47.57 $\pm$ 0.52        | 89.21 $\pm$ 0.20     |
| 0.8     | 82.89 $\pm$ 0.21        | 46.76 $\pm$ 0.47        | 89.03 $\pm$ 0.15     |
| 0.9     | 82.85 $\pm$ 0.23        | 46.49 $\pm$ 0.47        | 88.97 $\pm$ 0.20     |