Permutation Entropy for Graph Signals: Theoretical framework and application to neuroimaging data

In the field of time series analysis, the application of entropy metrics, particularly permutation entropy, is pivotal for quantifying system state probabilities and the complexity of signals. While these metrics have been extensively applied to uni-variate time series, and recently extended to two-dimensional data like images, their application to irregular domains such as graphs has remained unexplored. Here, we present a ground-breaking shift in this field by generalizing the concept of permutation entropy for the analysis of signals on irregular graphs. We have expanded the scope of permutation entropy, employing a method that compares signal values across neighbouring nodes using the graph's adjacency matrix. This generalization is not merely an adaptation but a significant enhancement of the classical permutation entropy applied to time series, extending its applicability. Our approach integrates signal values with the graph topology and provides a unifying framework for the analysis of uni- and multi-variate time series, images, and graph signals. This advancement has shown promising initial results including industrial applications and notably MRI brain signal analysis in the context of early Alzheimer’s disease, offering a new perspective and tools for analysing complex systems data.