Abstract Algebraic Tools for Systems Analysis

Introduction

The study of complex systems is a broad field involving researchers from a variety of backgrounds—economics, ecology, biology, physics, sociology, etc. Within this diversity of fields, a common methodological approach is to use numerical techniques and simulations to analyze large-scale network models. A well-known issue is that while these techniques are able to provide valuable examples of possible outcomes, they do little to provide insight into a general theoretical understanding of the nature of dynamical network processes. What is needed is a method of determining the qualitatively different behaviors of the system and an understanding of why they occur. However, the standard mathematical tools are unable to provide analytic solutions to only the simplest cases. The goal of this project is to introduce a different mathematical approach to address these issues, and to work towards providing a more general understanding of what can or cannot occur in a complex system.

Methods

The mathematical techniques come mainly from abstract algebra, including representation theory and Lie groups. While this is a deep branch of mathematics with applications to a variety of pure and applied fields, the methods have yet to be fully adapted to complex systems. Some initial work in this area has been done, e.g., [1], which provides a useful method of finding generic behavior of dynamic network processes. Furthermore, the use of small-scale modeling becomes important. By providing tools to understand the behavior of smaller models, it then becomes possible to understand the issues that arise when the networks grow in size.                                          

Preliminary results

Prior to joining IIASA in September 2014, I had applied these techniques to uniquely characterize the mathematical structure of strategic interaction, and this work is continuing to develop by incorporating a social structure to agents' behaviors. Furthermore, results on the mean-field approximation to dynamical network processes is providing insights into how the different network structures can affect observable model behavior.

Last edited: 02 March 2016