Persistent Entropy as a Detector of Phase Transitions

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.

Figures (10)

• Figure 1: Kuramoto Network with 50 oscillators. The network layout is produced with the Fruchterman Reingold plugin of Gephi and the colour of each node is proportional to the degree bastian2009gephi. The network is one example of the possible realizations.
• Figure 2: Kuramoto order parameter $r(t)$ for increasing coupling values $K$. Solid lines represent the mean across realizations, while shaded regions indicate one standard deviation. A sharp transition from incoherent dynamics to full synchronization is observed as $K$ increases.
• Figure 3: Temporal evolution of the normalized persistent entropy $\mathrm{NPE}(H_0)(t)$ for the Kuramoto model across different coupling strengths $K$. For low coupling, persistent entropy fluctuates around a high value, while for sufficiently large $K$ it rapidly drops and stabilizes, indicating a collapse of topological complexity.
• Figure 4: Persistence barcodes for the Kuramoto model at three representative time points ($t=1.0$, $t=2.0$, $t=3.0$) for coupling strength $K=5$. Left panels show $H_0$ barcodes, while right panels show $H_1$ barcodes. The collapse of $H_0$ bars and the absence of persistent $H_1$ features indicate convergence toward a fully synchronized configuration.
• Figure 5: Probability $\mathbb{P}(t_\ast(K)\leq T_{\max})$ of reaching topological stability within the observation horizon as a function of the coupling strength $K$. The horizontal dashed line denotes the confidence threshold $p_0$, while the vertical dashed line marks the estimated critical coupling $\hat{K}_c$.

Theorems & Definitions (8)

• Definition 1: Persistent Entropy
• Theorem 1: Persistent entropy detects phase transitions
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Corollary 1