We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, i.e., graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to consider points on the graph which are vertices or points on an edge. Our covariance functions are defined on the vertices and edge points of these graphs and are isotropic in the sense that they depend only on the geodesic distance or on a new metric called the resistance metric (which extends the classical resistance metric developed in electrical network theory on the vertices of a graph to the continuum of edge points). We discuss the advantages of using the resistance metric in comparison with the geodesic metric as well as the restrictions these metrics impose on the investigated covariance functions. In particular, many of the commonly used isotropic covariance functions in the spatial statistics literature (the power exponential, Mat{e}rn, generalized Cauchy, and Dagum classes) are shown to be valid with respect to the resistance metric for any graph with Euclidean edges, whilst they are only valid with respect to the geodesic metric in more special cases.

中文翻译：

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图及其边上的各向同性协方差函数

我们在线性网络上开发了协方差函数的参数类，并将它们扩展到具有欧几里得边的图，即边被视为线段的图或更一般的具有坐标系的集合，允许我们考虑图上的顶点或点一个边缘。我们的协方差函数定义在这些图的顶点和边点上，并且是各向同性的，因为它们仅取决于测地距离或称为电阻度量的新度量（它扩展了电气网络理论中开发的经典电阻度量）图的顶点到边缘点的连续体）。我们讨论了使用电阻度量与测地线度量相比的优势，以及这些度量对研究的协方差函数施加的限制。