Graph-Based Topological Entropy Quantification for Assessing Complexity Across Diverse Systems

The structural configuration of a material is a critical determinant of its physical and chemical properties. By manipulating structural properties, researchers can engineer materials with desired characteristics. High-entropy alloys (HEAs) exemplify this principle, as their properties are governed by the disordered atomic arrangement within a solid solution, rather than a uniform ordered phase. Similarly, physically unclonable functions (PUFs) leverage the inherent physical complexity of materials to hash input signals, with performance directly linked to material complexity. Given the importance of structural complexity in material design, its quantification becomes paramount. Traditionally, thermodynamic configurational entropy has been employed to measure this complexity. However, this approach is limited to systems with rapid dynamics due to its requirement for extensive data sampling. To address these limitations, a novel method based on information theory and graph isomorphism was proposed. While this approach has shown efficacy for network-forming materials such as silica, its applicability to diverse material structures remains unproven. This study introduces a methodology for quantifying the configurational entropy of the general systems, utilizing graph isomorphism theory. Comparative analysis of thermodynamic configurational entropy and the proposed graph entropy for arbitrary systems revealed consistent trends between the two measures. Furthermore, it was also revealed that the proposed configuration entropy can be generalized to work on systems other than silicon networks, such as triangular lattices, square lattices, and even amorphous materials. These findings have significant implications for designing and optimizing high-entropy materials and PUFs, whose physical properties are largely dependent on structural complexity. The proposed methodology offers a more efficient and versatile approach to quantifying structural complexity across a broader range of material systems.