For many important network types, physical coordinate systems and physical distances are either difficult to discern or inapplicable. Accordingly, coordinate systems and characterizations based on hop-distance measurements, such as Topology Preserving Maps (TPMs) and Virtual-Coordinate (VC) systems are attractive alternatives to geographic coordinates for many network algorithms. We present an approach to recover geometric and topological properties of a network with a small set of distance measurements. The approach is a combination of shortest path (often called geodesic) recovery concepts and low-rank matrix completion, generalized to the case of hop-distances in graphs. Results for sensor networks embedded in 2-D and 3-D spaces as well as for social networks indicate that the method can accurately capture the network connectivity with a small set of measurements. TPM generation can now be based on various context appropriate measurements or VC systems instead of distances to a set of global anchors. The proposed method is a significant generalization that allows the topology to be extracted from a random set of graph shortest paths, making it applicable in contexts such as social networks where VC generation may not be possible.

中文翻译：

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部分虚拟坐标和图测地线的网络拓扑映射

对于许多重要的网络类型，物理坐标系和物理距离很难分辨或不适用。因此，对于许多网络算法而言，基于跳距测量的坐标系和特征（例如拓扑保留图（TPM）和虚拟坐标（VC）系统）是地理坐标的有吸引力的替代方案。我们提出了一种使用少量距离测量来恢复网络的几何和拓扑特性的方法。该方法是最短路径（通常称为测地线）恢复概念和低秩矩阵完成度的组合，通常适用于图中跳距的情况。嵌入在2-D和3-D空间中的传感器网络以及社交网络的结果表明，该方法可以通过少量测量即可准确捕获网络连接。TPM生成现在可以基于各种上下文相关的测量或VC系统，而不是到一组全局锚点的距离。所提出的方法是一种重要的概括，它允许从图最短路径的随机集合中提取拓扑，使其适用于可能无法生成VC的环境，例如社交网络。