Decoding collective dynamics and complexity in nanoparticle assemblies using graph theory

Being intermediate in scale between molecules and colloids, nanoparticles combine characteristics of both. The structure of their self-assembled states combining order and disorder is difficult to quantify using traditional symmetry-based descriptors. Here, we applied graph theory (GT) to analyze assemblies of 400 to 10,000 nanoparticles across three material systems. We show that GT metrics, augmented Forman-Ricci curvature (AFRC) and Ollivier-Ricci curvature (ORC), capture local and global structural transitions from small clusters to extended networks. AFRC reflects the energetic state of the assembly, whereas ORC quantifies structural complexity and reveals a "Goldilocks" regime that maximizes plasmonic response. The generality of this approach is demonstrated for gold nanocubes, gold nanoprisms, and indium tin oxide nanospheres, providing a unified framework for describing and optimizing complex nanoparticle assemblies.