Mathematics

Domain (mathematical analysis)

Pipeline (software)

Politics

Data mining

Variety (cybernetics)

Mathematical analysis

Philosophy

Political science

Artificial intelligence

Relation (database)

Set (abstract data type)

Programming language

Data science

Representation (politics)

Vocabulary

Computer science

Linguistics

Complex system

Interpretation (philosophy)

Theoretical computer science

SIAM Review

Complex systems, composed at the most basic level of units and their interactions, describe phenomena in a wide variety of domains, from neuroscience to computer science and economics. The wide variety of applications has resulted in two key challenges: the generation of many domain-specific strategies for complex systems analyses that are seldom revisited, and the compartmentalization of representation and analysis ideas within a domain due to inconsistency in complex systems language. In this work we propose basic, domain-agnostic language in order to advance toward a more cohesive vocabulary. We use this language to evaluate each step of the complex systems analysis pipeline, beginning with the system under study and data collected, then moving through different mathematical frameworks for encoding the observed data (i.e., graphs, simplicial complexes, and hypergraphs), and relevant computational methods for each framework. At each step we consider different types of dependencies; these are properties of the system that describe how the existence of an interaction among a set of units in a system may affect the possibility of the existence of another relation. We discuss how dependencies may arise and how they may alter the interpretation of results or the entirety of the analysis pipeline. We close with two real-world examples using coauthorship data and email communications data that illustrate how the system under study, the dependencies therein, the research question, and the choice of mathematical representation influence the results. We hope this work can serve as an opportunity for reflection for experienced complex systems scientists, as well as an introductory resource for new researchers.

The Why, How, and When of Representations for Complex Systems