Survey and Review

Abstract

Our Survey and Review paper, “The Why, How, and When of Representations for Complex Systems,” by Leo Torres, Ann S. Blevins, Danielle Bassett, and Tina Eliassi-Rad, lists 233 references. Some of them were published in applied mathematics journals, like SIAM Review or SIAM Journal on Applied Algebra and Geometry. Others appeared in well-known “pure” mathematics journals, including Annals of Mathematics and Bulletin of the American Mathematical Society. There are many references to multidisciplinary outlets like Science, Nature, or Proceedings of the National Academy of Sciences of the USA. But the whole list is astonishing: from the Journal of the Acoustical Society of America to Sociometry to Human Brain Mapping. This diversity of journals is obviously due to the interest in complex systems that now exists in a wide variety of applications. A complex system is a collection of a large number of objects or agents that interact with one another in a way that gives rise to the emergence of collective behaviors not immediately predictable from the aggregation of the individual behaviors of the parts. The generality of this definition---the objects may be anything from molecules in a cell to businesses in the economy of a country---makes it possible for complex system theory to be useful in a large variety of fields, as described above. On the flip side, a theory that is spread over such a diversity of application areas may easily be compartmentalized. The aim of the review in this issue is to propose a common language to increase the cohesion of the theory. Three alternative approaches to the representation of complex systems are discussed in detail: graphs, simplicial complexes, and hypergraphs. The main ideas are illustrated by means of more than a dozen figures, and the paper closes with two real-world examples using coauthorship data and email communication data. In addition to being useful for complex system scientists wishing to broaden their view of the theory, the survey provides an accessible introduction to the field.