Permutation Entropy for Time Series Over n-Distance Path Graphs

Permutation entropy (PE) is a widely used metric for quantifying the complexity of time series data. Recent efforts have extended PE to graph signals, resulting in the graph permutation entropy (GPE), which accounts for the interdependencies encoded in the graph structure underlying the signal. Despite its potential, GPE remains a relatively recent development, lacking systematic formulations and practical validations. In this paper, we revisit the definition of GPE by introducing a formulation based on the graph shift operator (GSO), which enables a more natural interpretation and facilitates the construction of graph-based versions of other PE variants. As an example, we define the fractional graph permutation entropy (FrGPE), a generalization of GPE that incorporates a tunable fractional order parameter. We then investigate the use of GPE and FrGPE for analyzing time series defined over n-distance path graphs. Through computational experiments, we demonstrate the effectiveness of the proposed entropies in two scenarios: 1) detecting stability islands and distinguishing chaotic sequences generated with close parameters and 2) classifying faults in rolling bearings, where classification accuracies above 99% were achieved. These results highlight the potential of the proposed entropies as powerful tools for analyzing structured time series data.