The Straightness is a measure designed to characterize a pair of vertices in a spatial graph. It is defined as the ratio of the Euclidean distance to the graph distance between these vertices. It is often used as an average, for instance to describe the accessibility of a single vertex relatively to all the other vertices in the graph, or even to summarize the graph as a whole. In some cases, one needs to process the Straightness between not only vertices, but also any other points constituting the graph of interest. Suppose for instance that our graph represents a road network and we do not want to limit ourselves to crossroad-to-crossroad itineraries, but allow any street number to be a starting point or destination. In this situation, the standard approach consists in: 1) discretizing the graph edges, 2) processing the vertex-to-vertex Straightness considering the additional vertices resulting from this discretization, and 3) performing the appropriate average on the obtained values. However, this discrete approximation can be computationally expensive on large graphs, and its precision has not been clearly assessed. In this article, we adopt a continuous approach to average the Straightness over the edges of spatial graphs. This allows us to derive 5 distinct measures able to characterize precisely the accessibility of the whole graph, as well as individual vertices and edges. Our method is generic and could be applied to other measures designed for spatial graphs. We perform an experimental evaluation of our continuous average Straightness measures, and show how they behave differently from the traditional vertex-to-vertex ones. Moreover, we also study their discrete approximations, and show that our approach is globally less demanding in terms of both processing time and memory usage. Our R source code is publicly available under an open source license.

翻译：直线度是一种衡量空间图中两顶点之间关系的度量，定义为欧氏距离与图距离的比值。它常用作平均值，例如描述某个顶点相对于图中所有其他顶点的可达性，甚至用于总结整个图的特征。在某些情况下，我们需要处理的不仅限于顶点间的直线度，还包括构成关注图的任意点之间的直线度。例如，假设图代表道路网络，我们并不希望仅限于十字路口到十字路口的路径，而是允许任何街道号码作为起点或终点。在此情形下，标准方法包括：1) 将图边离散化，2) 考虑离散化引入的额外顶点，计算顶点到顶点的直线度，以及3) 对所得值进行相应的平均计算。然而，这种离散近似在大规模图上计算开销较大，且其精度尚未得到明确评估。本文采用连续方法对空间图边上的直线度进行平均。由此，我们推导出5种不同的度量，能够精确表征整个图以及单个顶点和边的可达性。我们的方法具有通用性，也可应用于其他为空间图设计的度量。我们对所提出的连续平均直线度度量进行了实验评估，展示了它们与传统顶点到顶点度量的不同表现。此外，我们还研究了它们的离散近似，并表明我们的方法在处理时间和内存使用方面总体要求更低。我们提供的R语言源代码已根据开源许可证公开可用。