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For many important network types (e.g., sensor networks in complex harshenvironments and social networks) physical coordinate systems (e.g.,Cartesian), and physical distances (e.g., Euclidean), are either difficult todiscern or inapplicable. Accordingly, coordinate systems and characterizationsbased on hop-distance measurements, such as Topology Preserving Maps (TPMs) andVirtual-Coordinate (VC) systems are attractive alternatives to Cartesiancoordinates for many network algorithms. Herein, we present an approach torecover geometric and topological properties of a network with a small set ofdistance measurements. In particular, our approach is a combination of shortestpath (often called geodesic) recovery concepts and low-rank matrix completion,generalized to the case of hop-distances in graphs. Results for sensor networksembedded in 2-D and 3-D spaces, as well as a social networks, indicates thatthe method can accurately capture the network connectivity with a small set ofmeasurements. TPM generation can now also be based on various contextappropriate measurements or VC systems, as long as they characterize differentnodes by distances to small sets of random nodes (instead of a set of globalanchors). The proposed method is a significant generalization that allows thetopology to be extracted from a random set of graph shortest paths, making itapplicable in contexts such as social networks where VC generation may not bepossible.

Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics