标题： 表面网络 显示英文标题

标题： Surfacic networks

摘要： 表面网络是建立在二维流形上的结构。 许多系统，包括交通网络和各种城市网络，都属于这一类。 节点高程的波动意味着与典型平面网络有显著偏差，需要特定的工具来理解其影响。 在这里，我们提出了这样的工具，包括最小化高程差异的懒惰路径、衡量最短路径疲劳程度的图难度，以及表征最短路径上正高程变化的额外努力。 我们使用表面网络的玩具模型来说明这些度量，并实证研究了选定城市中的行人网络。 具体来说，我们研究了高程变化如何影响介数中心性的空间分布。 我们还证明了额外努力遵循非平凡的幂律分布，其指数并不具有普遍性，这表明无论起点和终点之间的高程差如何，沿着最短路径遇到陡坡的概率都是显著的。 这些发现突显了高程波动在网络特性形成中的重要性。 表面网络为全面分析和建模位于或受限于表面环境的复杂系统提供了一个有前景的框架。 显示英文摘要

摘要： Surfacic networks are structures built upon a two-dimensional manifold. Many systems, including transportation networks and various urban networks, fall into this category. The fluctuations of node elevations imply significant deviations from typical plane networks and require specific tools to understand their impact. Here, we present such tools, including lazy paths that minimize elevation differences, graph arduousness which measures the tiring nature of shortest paths, and the excess effort, which characterizes positive elevation variations along shortest paths. We illustrate these measures using toy models of surfacic networks and empirically examine pedestrian networks in selected cities. Specifically, we examine how changes in elevation affect the spatial distribution of betweenness centrality. We also demonstrate that the excess effort follows a non-trivial power law distribution, with an exponent that is not universal, which illustrates that there is a significant probability of encountering steep slopes along shortest paths, regardless of the elevation difference between the starting point and the destination. These findings highlight the significance of elevation fluctuations in shaping network characteristics. Surfacic networks offer a promising framework for comprehensively analyzing and modeling complex systems that are situated on or constrained to a surface environment.