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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. [ABSTRACT FROM AUTHOR]