Sample entropy for graph signals: An approach to nonlinear analysis of graph signals

TL;DR

This paper introduces SampEn$_{G}$, a novel graph-signal generalization of Sample Entropy, enabling nonlinear analysis of graph signals by capturing irregularity through multi-hop neighborhood patterns.

Contribution

It extends classical SampEn to graph signals using a multi-hop embedding approach, validated on various graph types and demonstrating nonlinear sensitivity.

Findings

SampEn$_{G}$ reduces to classical SampEn on path graphs.

Validated nonlinear sensitivity with the logistic map.

Practical runtime on thousands of nodes.

Abstract

We introduce a graph-signal generalisation of Sample Entropy, denoted SampEn$_{G}$, to quantify irregularity of graph signals on a continuous state space, complementing existing methods on symbolic dynamics. Our approach replaces the temporal delay embedding of classical SampEn with a multi-hop graph-based embedding: for each node, we aggregate patterns from local walk-weighted neighbourhood averages computed via powers of the graph shift operator. We show empirically that SampEn$_{G}$ reduces to classical 1D SampEn on directed path graphs, and validate its nonlinear sensitivity using the logistic map. Experiments on directed Erd\H{o}s--R\'enyi graph signals further characterise its behaviour with connectivity and pattern length $m$, with practical runtimes on the order of thousands of nodes. We expect SampEn$_{G}$ to open up new ways to analyse graph signals, generalising SampEn and the concept of conditional entropy to extending nonlinear analysis to a wide variety of network data.