Comparing Structure and Dynamics of Transition Graphs by the Symmetric Difference Metric Over an Edge-Filtration

Transition graphs or transition diagrams, describing the rates and probabilities with which a system changes between discrete states, are common throughout the sciences. In many cases, parameterisations of transition graphs are inferred from different datasets, for example in the context of Markov or hidden Markov models. An important task for followup analysis is to find efficient and effective ways to compare transition graphs with different parameterisations. Here, we introduce the Weight-Filtration Comparison Curve (WFCC), an approach by which the differences between two or more parameterisations of a transition graph can be quantified and compared. Borrowing from topological data analysis, the WFCC allows graphs learned from different datasets and/or null models to be systematically compared, and differences in both the fine- and coarse-grained structure and dynamics of transition graphs to be quantitatively assessed. We demonstrate WFCC with simple illustrative cases and real-world cases of transition graphs inferred from global data on the evolution of antimicrobial resistance in different countries, showing how different inferred dynamics, and different levels of uncertainty, are reported by structural aspects of these comparison curves.