Information topology identifies emergent model classes

We introduce the concept of information topology by considering the information geometry of models of the same physical system under different experimental conditions. By varying the experimental conditions, statistical inference imposes different metrics on the system's parameter space, leading to different geometric properties, such as distance and curvature. Experimental conditions that preserve the structural identifiability of the parameters are related by diffeomorphisms and form an information topology. Experimental conditions that lead to a manifold collapse correspond to a different topology and require a modification of the underlying theory in order to construct a model whose parameters are identifiable (either structurally or practically). For many models the relevant topological feature is a hierarchical structure of boundaries (faces, edges, corners, etc.) which we represent as a hierarchical graph. The tips of this hierarchical graph correspond to dominant modes that govern the macroscopic, collective behavior of the system. We refer to these modes as emergent model classes. When new parameters are added to the model, we consider how the simple, original topology is embedded in the new, larger parameter space. When the emergent model classes are unstable to the introduction of new parameters, we classify the new parameters as relevant. In contrast, model classes are stable to the introduction of irrelevant parameters. In this way, information topology provides a unified mathematical language for understanding the relationship between experimental data, mathematical models, and the underlying theories from which they are derived and can provide a concrete framework for identifying an appropriate representation of a physical system.