Hierarchy of topological transitions in a network liquid.

The representation of complex systems as networks has become a critical tool across many fields of science. In the context of physical networks, such as biological neural networks, vascular networks, or network liquids where the nodes and edges occupy volume in three-dimensional space, the question of how they become densely packed is of special importance. Here, we investigate a model network liquid, which is known to densify via two successive liquid-liquid phase transitions (LLPTs). We elucidate the importance of rings-cyclic paths formed by bonded particles in the networks-and their spatial disposition in understanding the structural changes that underpin the increase in density across the LLPTs. Our analyses demonstrate that the densification of these networks is primarily driven by the formation of linked rings, and the LLPTs correspond to a hierarchy of topological transitions where rings form the fundamental building blocks. We envisage entanglement to emerge as a general mechanism for densification, with wide implications for the embedding of physical networks, especially in confined spaces.