  Rings comprising chemically bonded atoms are essential topological motifs for
the structural ordering of network-forming materials. Quantification of such
larger motifs beyond short-range pair correlation is essential for
understanding the linkages between the orderings and macroscopic behaviors.
Here, we propose two quantitative analysis methods based on rings. The first
method quantifies rings by two geometric indicators: roundness and roughness.
These indicators reveal the linkages between highly symmetric rings and crystal
symmetry in silica and that the structure of amorphous silica mainly consists
of distorted rings. The second method quantifies a spatial correlation function
that describes three-dimensional atomic densities around rings. A comparative
analysis among the functions for different degrees of ring symmetries reveals
that symmetric rings contribute to the local structural order in amorphous
silica. Another analysis of amorphous models with different orderings reveals
anisotropy of the local structural ordering around rings; this contributes to
building the intermediate-range ordering.
