Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics

TL;DR

This paper introduces a novel method combining low-rank matrix completion and geodesic recovery to accurately map network topology using partial virtual coordinates, applicable even with significant missing data.

Contribution

It presents a generalized approach for topology mapping from partial virtual coordinates and graph geodesics, extending applicability to networks lacking global anchor points.

Findings

Accurate topology maps are achievable with 40-60% missing virtual coordinates.

Method works with different types of virtual coordinate systems.

Topology can be recovered from random graph geodesics, broadening application scope.

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 inappropriate. Accordingly, Topology Preserving Maps (TPMs) derived from a Virtual-Coordinate (VC) system representing the distance to a small set of anchors is an attractive alternative to physical coordinates for many network algorithms. Herein, we present an approach, based on the theory of low-rank matrix completion, to recover geometric properties of a network with only partial information about the VCs of nodes. In particular, our approach is a combination of geodesic recovery concepts and low-rank matrix completion, generalized to the case of hop-distances in graphs. Distortion evaluated using the change of distance among node pairs shows that even with up to 40% to 60% of random coordinates missing, accurate TPMs can be obtained. TPM generation can now also be based on different context appropriate VC systems or measurements as long as they characterize each node with distances to a small set of random nodes (instead of a global set of anchors). The proposed method is a significant generalization that allows the topology to be extracted from a random set of graph geodesics, making it applicable in contexts such as social networks where VC generation may not be possible.