Topological similarity of random cell complexes and applications

Area of Science:

• Physical Sciences
• Materials Science
• Computational Topology

Background:

• Random cell complexes are prevalent in physical sciences but lack standardized methods for statistical comparison.
• Existing methods are often system-specific and not universally applicable.
• A general framework is needed to analyze and compare the local topology of cell complexes.

Purpose of the Study:

• To develop a general and complete method for quantifying statistical similarities and differences in cell complexes.
• To introduce novel concepts, the swatch and cloth, for describing local cell complex topology.
• To define a distance metric for comparing the local topology of two cell complexes.

Main Methods:

• Introduction of the swatch and cloth as general descriptors of local cell complex topology.
• Development of a distance metric based on the swatch and cloth to measure topological similarity.
• Application of the distance metric to analyze model grain boundary and dislocation networks.

Main Results:

• The swatch and cloth provide a general and complete description of local cell complex topology.
• A novel distance metric effectively quantifies topological similarity between cell complexes.
• The distance metric was used to identify a steady state in a grain boundary network, independent of initial conditions.
• The metric demonstrated that long-term properties of a dislocation network model are independent of intersection implementation.

Conclusions:

• The swatch and cloth offer a universally applicable framework for analyzing cell complex topology.
• The defined distance metric is a powerful tool for comparing complex systems in physical sciences.
• This approach facilitates the study of system dynamics, steady states, and initial condition independence.