For a young science like complexity, a good textbook marks a developmental milestone. It draws together relevant insights, organizes them in a central framework, and presents the methods of research by which new scientists can enter the field.

Physics has its “Feynman Lectures.” Biology has a proliferation of eponymous reference textbooks. But their offspring, complexity science, has notably lacked a touchstone text. Until now.

This month, Oxford University Press published Introduction to the Theory of Complex Systems by SFI External Professor Stefan Thurner, Rudolf Hanel, and Peter Klimek, all based at Complexity Science Hub Vienna. Their book synthesizes hundreds of disparate findings in complexity and articulates a single, underlying characteristic of complex systems: They work like co-evolving algorithms.

In other words, the kaleidoscope of complex systems — ant colonies, economies, social networks, spin glasses —  are best described by the rules that govern their interactions, rather than just the properties of the individual components. As individual ants, banks, friends, or molecules interact with each other and adapt accordingly, the states of the components change which in turn causes the nature and strength of the interactions themselves to change, in what Thurner describes as “a chicken-and-egg problem.”

“If you think of a system being in one particular state, then you update it in steps. First, you update the states then you update the interactions then the states, etc. — that’s an algorithm where interactions and states co-evolve,” Thurner explains.

He cites the healthcare system as an example. When a patient sees a doctor, that’s one type of interaction. When they go to the pharmacy to pick up a prescription, that’s another. Viewing the healthcare system through a complex systems lens involves examining a big dataset of these interactions at regular intervals.

“For example we look at the healthcare system every six months or so, using data from, say, an insurance company,” Thurner says. “We look at how everyone is doing at one point in time – the “health state” of everyone… and we see how patients “interact with the doctors.” And we see how diseases of individuals progress as the result of these interactions. We see how the whole healthcare system updates itself at various states. As people get healthier, they tend to interact less with the system. As a result, we can learn for example what approaches are working for whom, and for which diseases.”

The short history of complexity

This co-evolutionary algorithmic approach departs from the analytic methods that have characterized mainstream science since the time of Isaac Newton. Instead of ‘zooming in’ on the properties and characteristics of individual planets, rocks, and atoms, complexity science seeks to uncover emergent, system-wide behaviors that can’t be predicted simply by knowing more details about the individual components.

Founded in 1984, the Santa Fe Institute became the first research institute dedicated to investigating and discovering the new principles of complex systems.  Since then, SFI scientists have helped pioneer new techniques like network science, statistical mechanics, evolutionary processes, and biological scaling theory, all of which are explored in the new textbook.

“All of the roots of this book are in Santa Fe,” Thurner says, “including Murray Gell-Mann’s work on what a complex system is.”

In the book’s introduction, Thurner, Hanel, and Klimek compare the modern state of complexity science to that of quantum mechanics in the 1920s, before the Copenhagen meetings and the publication of Werner Heisenberg’s The Physical Principles of the Quantum Theory.  

“At that time, quantum mechanics was a collection of experimental and theoretical bits and pieces, which had not yet been seen within a fully comprehensive framework,” they write. “Nevertheless, it was clear that, one day soon, such a framework would exist.”

Introduction to a Theory of Complex Systems does for complexity what Heisenberg’s book did for quantum mechanics. It synthesizes the insightful bits and pieces from isolated research papers, and it also presents something new:  an underlying algorithmic theory that proposes to unify previously separate observations in complexity science.

A theory of complex adaptive systems

“Complexity until now has been lacking a strong theoretical underpinning. Now it has one,” External Professor W. Brian Arthur says of the new textbook. Arthur is one of the founders of the Santa Fe Institute and a complexity science pioneer.

The theory the authors present is a novel, universal and coherent mechanism — a co-evolutionary algorithmic view of complex systems — that explains the origins of many of the statistical patterns that have been observed in complex systems across many domains. The algorithm describes a random process, like dice-rolling, that can generate anything from Gaussian bell curves to the “fat-tailed” distributions that describe earthquakes and financial crashes.

Using their theory on non-equilibrium complex processes, Thurner says that a few observations about a complex system will tell scientists which kind of statistics they can expect to observe. These in turn allow them to compute systemic properties of the systems.

As for who will benefit from the book, the authors have two main audiences in mind. The narrower group: late-undergraduate to graduate students in physics and mathematics. More broadly, it’s for complexity enthusiasts who want to develop a deeper understanding of the field, beyond verbal descriptions.

According to Distinguished Professor Geoffrey West, “This book will surely become a standard text for anyone who wants to seriously understand complexity no matter what their background or stage of career. It is written from a physicist’s perspective, stressing mechanism, underlying principles and mathematical rigour, yet is eminently readable and pedagogical.”

The hardcover is available through Oxford University Press.

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