Structural and Compositional Complexities of Hierarchical Self-Assembly: a Hypergraph Approach

The work argues that programmable self-assembly can build complex molecular, supramolecular, and crystalline structures from designed building blocks. It proposes a new way to represent these structures using a hypergraph formalism called Blocks & Bonds (B&B), which extends chemical graph ideas by allowing directed, multicolored interactions, internal symmetries, and hierarchical organization. To describe architectures, the authors introduce a language called Structure Code (SC). They then define two ways to measure how complex a structure is: a token-based, Kolmogorov-style Structural Complexity derived from SC information content, and a simpler Compositional Complexity based only on how many block and bond types are used and how often. Across the systems they examined, they report a strong empirical correlation between Structural Complexity and Compositional Complexity. They demonstrate the framework on molecular examples (ethylene glycol, glucose), DNA-origami lattices, and crystalline assemblies, claiming it provides a unified, scalable representation that captures symmetry, modularity, and stereochemistry, and supports complexity-aware classification and inverse design of programmable matter.