Permutation Entropy for Graph Signals

TL;DR

This paper introduces a novel permutation entropy metric for analyzing signals on irregular graph domains, extending nonlinear analysis tools from time series and images to complex network structures.

Contribution

It develops the first permutation entropy method for graph signals, generalizing existing metrics to irregular domains using adjacency matrices.

Findings

Preserves properties of classical permutation entropy.

Applicable to synthetic and real graph signals.

Enables extension of nonlinear dynamic analysis to graphs.

Abstract

Entropy metrics (for example, permutation entropy) are nonlinear measures of irregularity in time series (one-dimensional data). Some of these entropy metrics can be generalised to data on periodic structures such as a grid or lattice pattern (two-dimensional data) using its symmetry, thus enabling their application to images. However, these metrics have not been developed for signals sampled on irregular domains, defined by a graph. Here, we define for the first time an entropy metric to analyse signals measured over irregular graphs by generalising permutation entropy, a well-established nonlinear metric based on the comparison of neighbouring values within patterns in a time series. Our algorithm is based on comparing signal values on neighbouring nodes, using the adjacency matrix. We show that this generalisation preserves the properties of classical permutation for time series and the recent permutation entropy for images, and it can be applied to any graph structure with synthetic and real signals. We expect the present work to enable the extension of other nonlinear dynamic approaches to graph signals.