Persistent Entropy as a Detector of Phase Transitions

TL;DR

This paper provides a theoretical foundation for using persistent entropy to detect phase transitions in complex systems, establishing conditions for reliable detection and validating the approach through diverse empirical examples.

Contribution

It introduces a general theorem ensuring persistent entropy's effectiveness in phase detection and develops a practical framework connecting theory with finite data analysis.

Findings

Persistent entropy reliably detects phase transitions across various systems.

Stabilization of persistent entropy indicates critical transition points.

Theoretical conditions ensure persistent entropy's separation of phases.

Abstract

Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.