We derive an explicit link between Gaussian Markov random fields on metric graphs and graphical models, and in particular show that a Markov random field restricted to the vertices of the graph is, under mild regularity conditions, a Gaussian graphical model with a distribution which is faithful to its pairwise independence graph, which coincides with the neighbor structure of the metric graph. This is used to show that there are no Gaussian random fields on general metric graphs which are both Markov and isotropic in some suitably regular metric on the graph, such as the geodesic or resistance metrics.

翻译：我们推导了度量图上的高斯马尔可夫随机场与图模型之间的显式关联，特别证明了在温和的正则性条件下，限制在图顶点上的马尔可夫随机场是一个高斯图模型，其分布忠实于其成对独立图，该图与度量图的邻域结构一致。基于此，我们证明了在一般度量图上不存在同时满足马尔可夫性且在图的某种适当正则度量（如测地线度量或电阻度量）下具有各向同性的高斯随机场。