Linear compartmental models (LCMs) have broad applications in fields like pharmacokinetics, epidemiology, ecology, and systems engineering, where they are used to describe the movement of substances or information between different compartments. A central challenge in the study of LCMs is identifying situations where models with distinct graphical structures can be indistinguishable. This study builds upon Dr. Bortner’s work by investigating the indistinguishability of directed cycle models, a type of LCM. Through a graph theoretical approach, we developed Python code to generate information about directed cycle LCMs, which we use to highlight the connections between model structures and their corresponding input-output equations. The primary objective is to uncover additional conditions for indistinguishability, enhancing the understanding of the structural and algebraic factors that contribute to it. The outcomes of this research are crucial for furthering the application of LCMs in various fields, especially in situations involving directed cycle models, providing valuable insights into their practical use and limitations.

Analyzing millions of patents and their classifications throughout time is a challenge for many researchers studying patent innovation. In this study, we create a reproducible metric to analyze innovation over time by observing the hierarchal changes in networks of classification throughout time. We also propose a new metric when looking at innovation throughout time by developing a framework for an evolutionary neural network. While our findings are primarily towards the field of firearm patents, this methodology can be generalized to other technological domains.