Although tremendous success has been achieved in spatial and network representation separately in recent years, there exist very little works on the representation of spatial networks. Extracting powerful representations from spatial networks requires the development of appropriate tools to uncover the pairing of both spatial and network information in the appearance of node permutation invariant, and rotation and translation invariant. Hence it can not be modeled merely with either spatial or network models individually. To address these challenges, this paper proposes a generic framework for spatial network representation learning. Specifically, a provably information-lossless and rotation-translation invariant representation of spatial information on networks is presented. Then a higher-order spatial network convolution operation that adapts to our proposed representation is introduced. To ensure efficiency, we also propose a new approach that relied on sampling random spanning trees to reduce the time and space complexity fromO(N3) to O(N).

Spatial networks are networks for which the nodes and edges are constrained by geometry and embedded in real space, which has crucial effects on their topological properties. Although tremendous success has been achieved in spatial and network representation separately in recent years, there exist very little works on the representation of spatial networks. Extracting powerful representations from spatial networks requires the development of appropriate tools to uncover the pairing of both spatial and network information in the appearance of node permutation invariant, and rotation and translation invariant. Hence it can not be modeled merely with either spatial or network models individually. To address these challenges, this paper proposes a generic framework for spatial network representation learning. Specifically, a provably information-lossless and roto-translation invariant representation of spatial information on networks is presented. Then a higher-order spatial network convolution operation that adapts to our proposed representation is introduced. To ensure efficiency, we also propose a new approach that relied on sampling random spanning trees to reduce the time and memory complexity from $O(N^3)$ to $O(N)$. We demonstrate the strength of our proposed framework through extensive experiments on both synthetic and real-world datasets.

Measuring similarity in urban spatial networks is key to understanding cities as complex systems. Yet most existing methods are not tailored for spatial networks and struggle to differentiate them effectively. We propose GCA-Sim, a similarity-evaluation framework based on graph cellular automata. Each submodel measures similarity by the divergence between value distributions recorded at multiple stages of an information evolution process. We find that some propagation rules magnify differences among network signals; we call this "network resonance." With an improved differentiable logic-gate network, we learn several submodels that induce network resonance. We evaluate similarity through clustering performance on fifty city-level and fifty district-level road networks. The submodels in this framework outperform existing methods, with Silhouette scores above 0.9. Using the best submodel, we further observe that planning-led street networks are less internally homogeneous than organically grown ones; morphological categories from different domains contribute with comparable importance; and degree, as a basic topological signal, becomes increasingly aligned with land value and related variables over iterations.

Vision-Language Models (VLMs) demonstrate remarkable potential in robotic manipulation, yet challenges persist in executing complex fine manipulation tasks with high speed and precision. While excelling at high-level planning, existing VLM methods struggle to guide robots through precise sequences of fine motor actions. To address this limitation, we introduce a progressive VLM planning algorithm that empowers robots to perform fast, precise, and error-correctable fine manipulation. Our method decomposes complex tasks into sub-actions and maintains three key data structures: task memory structure, 2D topology graphs, and 3D spatial networks, achieving high-precision spatial-semantic fusion. These three components collectively accumulate and store critical information throughout task execution, providing rich context for our task-oriented VLM interaction mechanism. This enables VLMs to dynamically adjust guidance based on real-time feedback, generating precise action plans and facilitating step-wise error correction. Experimental validation on complex assembly tasks demonstrates that our algorithm effectively guides robots to rapidly and precisely accomplish fine manipulation in challenging scenarios, significantly advancing robot intelligence for precision tasks.

Spatial networks are useful for modeling geographic phenomena where spatial interaction plays an important role. To analyze the spatial networks and their internal structures, graph-based methods such as community detection have been widely used. Community detection aims to extract strongly connected components from the network and reveal the hidden relationships between nodes, but they usually do not involve the attribute information. To consider edge-based interactions and node attributes together, this study proposed a family of GeoAI-enhanced unsupervised community detection methods called region2vec based on Graph Attention Networks (GAT) and Graph Convolutional Networks (GCN). The region2vec methods generate node neural embeddings based on attribute similarity, geographic adjacency and spatial interactions, and then extract network communities based on node embeddings using agglomerative clustering. The proposed GeoAI-based methods are compared with multiple baselines and perform the best when one wants to maximize node attribute similarity and spatial interaction intensity simultaneously within the spatial network communities. It is further applied in the shortage area delineation problem in public health and demonstrates its promise in regionalization problems.

Urban spaces, though often perceived as discrete communities, are shared by various functional and social groups. Our study introduces a graph-based physics-aware deep learning framework, illuminating the intricate overlapping nature inherent in urban communities. Through analysis of individual mobile phone positioning data at Twin Cities metro area (TCMA) in Minnesota, USA, our findings reveal that 95.7 % of urban functional complexity stems from the overlapping structure of communities during weekdays. Significantly, our research not only quantifies these overlaps but also reveals their compelling correlations with income and racial indicators, unraveling the complex segregation patterns in U.S. cities. As the first to elucidate the overlapping nature of urban communities, this work offers a unique geospatial perspective on looking at urban structures, highlighting the nuanced interplay of socioeconomic dynamics within cities.

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.

Geospatial sciences include a wide range of applications, from environmental monitoring transportation to infrastructure planning, as well as location-based analysis and services. Graph theory algorithms in mathematics have emerged as indispensable tools in these domains due to their capability to model and analyse spatial relationships efficiently. This article explores the applications of graph theory algorithms in geospatial sciences, highlighting their role in network analysis, spatial connectivity, geographic information systems, and various other spatial problem-solving scenarios like digital twin. The article provides a comprehensive idea about graph theory's key concepts and algorithms that assist the geospatial modelling processes and insights into real-world geospatial challenges and opportunities. It lists the extensive research, innovative technologies and methodologies implemented in this domain. Keywords: Graph theory, Geospatial Science, Digital twin 1. Introduction Geospatial science has developed as a vibrant field characterised by intellectual vigour, conceptual expansion, and improved analytical skills as a consequence of the Quantitative Revolution in the subject of geography through a spatially integrated socio-environmental science that outshines prior disciplinary ties, borders, and limitations (Berry et al., 2008). Geospatial science, commonly referred to as geomatics (Aina 2012), is a multidisciplinary discipline that focuses on comprehending, analysing, and visualising spatial data about the Earth's surface using information technology to describe the connections between geography, individuals, places, and Earth processes. Technologies like Global Positioning System (GPS), Geographic Information Systems (GIS), and remote sensing are frequently used as observational, measuring, and analytical tools, helping in the understanding of numerous events by providing the information with a spatial context. Geospatial technology is being used increasingly in every industry today, including resource management, disaster management, forestry, logistics, infrastructure planning, and the study of climate change and other environmental issues (Dangermond and Goodchild, 2020). Geospatial technology and the information created are becoming increasingly significant in all economic sectors, making the economy, society, and the environment an indispensable pillar of sustainable development. (Scott and Rajabifard, 2017).

Community Detection algorithms are used to detect densely connected components in complex networks and reveal underlying relationships among components. As a special type of networks, spatial networks are usually generated by the connections among geographic regions. Identifying the spatial network communities can help reveal the spatial interaction patterns, understand the hidden regional structures and support regional development decision-making. Given the recent development of Graph Convolutional Networks (GCN) and its powerful performance in identifying multi-scale spatial interactions, we proposed an unsupervised GCN-based community detection method "region2vec" on spatial networks. Our method first generates node embeddings for regions that share common attributes and have intense spatial interactions, and then applies clustering algorithms to detect communities based on their embedding similarity and spatial adjacency. Experimental results show that while existing methods trade off either attribute similarities or spatial interactions for one another, "region2vec" maintains a great balance between both and performs the best when one wants to maximize both attribute similarities and spatial interactions within communities.