We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous analog of the (discrete) mean distance is the mean of the distances between all pairs of points on the edges of the graph. Despite being a very natural generalization, to the best of our knowledge this concept has been barely studied, since the jump from discrete to continuous implies having to deal with an infinite number of distances, something that increases the difficulty of the parameter. In this paper we show that the continuous mean distance of a weighted graph can be computed in time quadratic in the number of edges, by two different methods that apply fundamental concepts in discrete algorithms and computational geometry. We also present structural results that allow a faster computation of this continuous parameter for several classes of weighted graphs. Finally, we study the relation between the (discrete) mean distance and its continuous counterpart, mainly focusing on the relevant question of the convergence when iteratively subdividing the edges of the weighted graph.

中文翻译：

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加权图的连续平均距离

我们研究加权图的连续平均距离的概念。对于连接的未加权图，平均距离可以定义为所有顶点对之间的距离的算术平均值。该参数提供了图的紧缩性的自然度量，并且已经对其进行了深入研究，包括几个变体，包括加权图的版本。（离散）平均距离的连续模拟是图形边缘上所有成对点之间的距离的平均值。尽管是非常自然的概括，但据我们所知，几乎没有研究过此概念，因为从离散到连续的跳跃意味着必须处理无限数量的距离，这增加了参数的难度。在本文中，我们表明，可以通过两种在离散算法和计算几何中应用基本概念的不同方法，在边数上以二次方的形式计算加权图的连续平均距离。我们还提出了结构结果，可以更快地计算几类加权图的该连续参数。最后，我们研究了（离散）平均距离与其连续对应距离之间的关系，主要集中在迭代地细分加权图的边缘时收敛的相关问题。我们还提出了结构结果，可以更快地计算几类加权图的该连续参数。最后，我们研究了（离散）平均距离与其连续对应距离之间的关系，主要集中在迭代地细分加权图的边缘时收敛的相关问题。我们还提出了结构结果，可以更快地计算几类加权图的该连续参数。最后，我们研究了（离散）平均距离与其连续对应距离之间的关系，主要集中在迭代地细分加权图的边缘时收敛的相关问题。