In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often  need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. But there is a lack of rigorous methods to compare different models a priori. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way. Using well-developed and understood concepts from simplicial geometry we are able to define a distance based on the persistent homology applied to the simplicial complexes that captures the model structure . In this way we can identify shared topological features of different models. We then expand this, and move from a distance between simplicial complexes to studying equivalences between models in order to determine their functional relatedness .
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often , therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way. Using well-developed concepts from simplicial algebraic topology, we are able to define a distance based on the persistent homology of the simplicial complexes . The persistent homology represents the structure of the models, so in this way we can identify shared topological features of different models. We then expand on this measure of distance between simplicial complexes to study the concept of equivalence between models in order to determine their conceptual similarity. We apply our methodology to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated .
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