Before revision:  Here, we demonstrate an unbiased machine learning approach to analyze multiparticle clusters based on topological structure, which is robust to changes in population size. For a given particle configuration , topological data analysis (TDA) determines the stability of spatial connectivity at varying length scales (i.e. persistent homology) , and can compare different particle configurations based on the "cost" of URLanizing one configuration into another.
After revision:  Classically, different phases and phase transitions have been described based on local ordering, which may not capture structural features at larger length scales. Instead , topological data analysis (TDA) determines the stability of spatial connectivity at varying length scales (i.e. persistent homology) , and can compare different particle configurations based on the "cost" of URLanizing one configuration into another.