Model comparison via simplicial complexes and persistent homology

Area of Science:

• Mathematical Modeling
• Computational Biology
• Topology

Background:

• Mathematical models are crucial in science and technology, but their structures are often poorly understood.
• Existing statistical methods compare models using data, but lack rigorous approaches for *a priori* model structure comparison.

Purpose of the Study:

• To develop and illustrate systematic methods for comparing mathematical model structures before data analysis.
• To define a quantifiable distance between models based on their topological representations.
• To establish a framework for assessing model equivalence and conceptual similarity.

Main Methods:

• Representing mathematical models as simplicial complexes.
• Applying concepts from simplicial algebraic topology to define a distance metric between models.
• Utilizing persistent homology and flat filtration for alternative model representations as persistence intervals.

Main Results:

• A novel distance measure between models is defined using their simplicial representations.
• Persistence intervals derived from persistent homology offer alternative representations of model structure.
• The methodology successfully demonstrated conceptual equivalence between a positional-information model and a Turing-pattern model.

Conclusions:

• The developed topological approach provides a rigorous framework for *a priori* mathematical model comparison.
• This method quantifies model distance and equivalence, revealing conceptual similarities.
• The findings highlight an unexpected equivalence in developmental biology models, previously considered unrelated.