Entropy as a key to the past: Quantifying adaptive cycles in complex networks: SUPPLEMENTARY MATERIAL

How can we objectively measure and characterize the phases of complex adaptive systems? Despite widespread recognition of the Adaptive Cycle Model's value for understanding system dynamics, its application has been constrained by the predominance of qualitative approaches. This paper introduces a novel methodological framework that channels the analytical power of entropy to quantify and characterize adaptive cycle phases in complex networks. Our approach proposes integrating four sophisticated entropy measures—degree, eigenvector, community, and betweenness entropy—into a comprehensive system for identifying and measuring phase transitions. We validate this innovative framework through an empirical test case analyzing archaeological networks from Eastern Iberia (5300-3800 cal BP), where entropy patterns reveal previously undetectable signatures of system transformation. The method's application demonstrates remarkable precision in phase identification, with entropy variations providing clear mathematical signatures for each adaptive cycle phase. This breakthrough in quantitative analysis enables objective comparison of system states. It reveals subtle patterns in phase transitions that traditional approaches miss entirely, opening new possibilities for studying complex system dynamics across multiple disciplines.