Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics

TL;DR

This paper introduces a novel method combining shortest path recovery and low-rank matrix completion to accurately map network topology from limited hop-distance measurements, applicable to sensor and social networks.

Contribution

It presents a generalized approach for topology mapping using partial virtual coordinates and graph geodesics, extending existing methods to networks without global anchors.

Findings

Accurately captures network connectivity with few measurements

Applicable to sensor and social networks in 2-D and 3-D spaces

Enables topology extraction from random shortest path sets

Abstract

For many important network types (e.g., sensor networks in complex harsh environments and social networks) physical coordinate systems (e.g., Cartesian), and physical distances (e.g., Euclidean), 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 Cartesian coordinates for many network algorithms. Herein, we present an approach to recover geometric and topological properties of a network with a small set of distance measurements. In particular, our 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 a social networks, indicates that the method can accurately capture the network connectivity with a small set of measurements. TPM generation can now also be based on various context appropriate measurements or VC systems, as long as they characterize different nodes by distances to small sets of random nodes (instead of 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.