A multiscale method for data collected from network edges via the line graph

TL;DR

This paper introduces a multiscale method using line graph wavelets for data analysis on network edges, addressing challenges in fields like hydrology where data naturally resides on edges rather than nodes.

Contribution

It develops a novel lifting scheme and wavelet transform on line graphs to enable multiscale data decomposition directly on network edges.

Findings

Effective denoising of water quality data in river networks.

Superior performance compared to existing node-based methods.

Validated through hydrology case studies and simulations.

Abstract

Data collected over networks can be modelled as noisy observations of an unknown function over the nodes of a graph or network structure, fully described by its nodes and their connections, the edges. In this context, function estimation has been proposed in the literature and typically makes use of the network topology such as relative node arrangement, often using given or artificially constructed node Euclidean coordinates. However, networks that arise in fields such as hydrology (for example, river networks) present features that challenge these established modelling setups since the target function may naturally live on edges (e.g., river flow) and/or the node-oriented modelling uses noisy edge data as weights. This work tackles these challenges and develops a novel lifting scheme along with its associated (second) generation wavelets that permit data decomposition across the network edges. The transform, which we refer to under the acronym LG-LOCAAT, makes use of a line graph construction that first maps the data in the line graph domain. We thoroughly investigate the proposed algorithm's properties and illustrate its performance versus existing methodologies. We conclude with an application pertaining to hydrology that involves the denoising of a water quality index over the England river network, backed up by a simulation study for a river flow dataset.