In this paper, we propose a methodology for deriving a model of a complex system by exploiting the information extracted from topological data analysis. Central to our approach is the S[B] paradigm in which a complex system is represented by a two-level model. One level, the structural S one, is derived using the newly-introduced quantitative concept of persistent entropy, and it is described by a persistent entropy automaton. The other level, the behavioral B one, is characterized by a network of interacting computational agents. The presented methodology is applied to a real case study, the idiotypic network of the mammalian immune system.

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.