In this paper we show how the coupling of the notion of a network with directions with the adaptation of the four-point probe from materials testing gives rise to a natural geometry on such networks. This four-point probe geometry shares many of the properties of hyperbolic geometry wherein the network directions take the place of the sphere at infinity, enabling a navigation of the network in terms of pairs of directions: the geodesic through a pair of points is oriented from one direction to another direction, the pair of which are uniquely determined. We illustrate this in the interesting example of the pages of Wikipedia devoted to Mathematics, or “The MathWiki.” The applicability of these ideas extends beyond Wikipedia to provide a natural framework for visual search and to prescribe a natural mode of navigation for any kind of “knowledge space” in which higher order concepts aggregate various instances of information. Other examples would include genre or author organization of cultural objects such as books, movies, documents or even merchandise in an online store.
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机译：在本文中，我们展示了网络概念与方向的耦合以及材料测试中四点探针的自适应如何在这种网络上产生自然的几何形状。这种四点探针几何形状具有双曲几何的许多特性，其中网络方向在无穷远处代替球体，从而使网络可以按成对的方向进行导航：通过一对点的测地线从一个方向到另一个方向，一对是唯一确定的。我们通过有趣的Wikipedia专用于数学的页面或“ The MathWiki”的示例来说明这一点。这些想法的适用范围超出了Wikipedia的范围，不仅为视觉搜索提供了自然的框架，而且为任何一种“知识空间”规定了一种自然的导航模式，在这种知识空间中，高阶概念聚集了各种信息实例。其他示例将包括文化对象（例如书籍，电影，文档甚至在线商店中的商品）的类型或作者组织。
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