Inverse problems in geometric graphs using internal measurements

TL;DR

This paper develops an algorithmic approach to determine the topology and geometry of lossy quantum graphs using internal measurements, leveraging algebraic and differential topology to improve upon traditional methods.

Contribution

It introduces a novel methodology combining algebraic and differential topology for inverse problems in quantum graphs, focusing on internal measurements and urban sensing applications.

Findings

Effective algorithms for reconstructing graph topology and geometry

Utilization of narrowband and visibility measurements

Separation of topology and geometry effects

Abstract

This article examines the inverse problem for a lossy quantum graph that is internally excited and sensed. In particular, we supply an algorithmic methodology for deducing the topology and geometric structure of the underlying metric graph. Our algorithms rely on narrowband and visibility measurements, and are therefore of considerable value to urban remote sensing applications. In contrast to the traditional methods in quantum graphs, we employ ideas related to algebraic and differential topology directly to our problem. This neatly exposes and separates the impact of the graph topology and geometry.