Traditional coursework makes ample use of the assumptions that systems are small, linear, and governed by Gaussian statistics, and for good reason: They make for tractable math. One could therefore forgive a STEM undergraduate for the belief that one can approximate most real systems with the simple harmonic oscillator, perhaps with a bit of white noise thrown in for good measure. Unfortunately, real systems typically violate all three assumptions, with dramatic consequences. Power blackouts, financial panics, and ecosystem collapses are just a few examples of problems that can be understood only in light of the large, the nonlinear, and the non-Gaussian. The ubiquity and outsize impact of such phenomena beg for broader awareness of the mathematics and consequences of complex systems. But how is that to be achieved without stripping away the attributes that make systems complex in the first place?
In An Introduction to Complex Systems: Society, Ecology, and Nonlinear Dynamics, Paul Fieguth takes on that challenging task. An engineer by training, he combines a practitioner’s sensibility with a hobbyist’s knowledge of a grab bag of fields ranging from climate change to soil science. His book represents a new take on the pedagogy of complex systems, emphasizing concepts and consequences over calculations.
The book reads like a play in three acts. The first portion, motivated by the timely problem of global warming, introduces the reader to systems theory. With the stage thus set, Fieguth tackles the question of how to define a system in the first place. What are the system’s boundaries? Is it open or closed? If open, what are its inputs and outputs, and what does thermodynamics imply about its operation? The book then deals with the representation of a system’s dynamics and associated dynamical state. Are the governing equations stationary in time? In space? What do we need to measure to make meaningful statements about the system’s behavior? The intent in posing those questions is not to provide one-size-fits-all answers, but to accustom the reader to thinking about the nuances of systems modeling.
The middle third of the text can be described as an atlas of complex systems, using the relatively bland linear systems as a counterpoint. Along the way, the author delves into new phenomena unlocked by each new layer of complexity; for instance, chapter 6 introduces nonlinearities in one-dimensional systems and with them the possibility of multi-stability, bifurcations, and hysteresis. Later chapters discuss higher-dimensional nonlinear systems and what the author terms “spatial systems”—those properly modeled by partial differential equations or agent-based models. Fieguth covers critical modeling issues like discretization, resolution, stability of numerical methods, and boundary-value constraints. That nod toward the fact that the science of complex systems is unavoidably computational gives the book a practical flavor that I like.
The final act covers behavior and problems common to most large complex systems. Chapter 9 shatters the illusion that bell curves dominate the statistics of real-world systems and uses examples as varied as drought lengths, city sizes, and movements of the Dow Jones Industrial Average to show that “extreme” and “improbable” are not synonymous. Next, readers are exposed to the key concept of emergent behavior, and chapter 11 provides a good introduction to the challenges of observation and inference in large complex systems. In a fitting coda, chapter 12 ties the themes together with an in-depth case study of water, from ocean acidification to groundwater availability.
Of course, one cannot treat many of the above topics without some appeal to the underlying mathematics. Fieguth strikes a good balance. He strips derivations to the bare essentials and sequesters the dry machinery of, say, eigendecomposition to an appendix. That frees him to focus on the interpretation of the problem at hand.
Occasionally, by emphasizing broad themes over nuances, Fieguth does a disservice to his readers. Take, for instance, the focus on power-law distributions, which Fieguth claims are the ubiquitous face of non-Gaussian statistics across nature and society. In fact, many reported power laws lack statistical support or a plausible generative mechanism; the author would have done better to emphasize the contrast between thin-tailed and heavy-tailed distributions in general. Conversely, at times the text pursues tangents at the expense of the main message. A cursory, few-page excursion into control theory, for example, ends up being more jarring than illuminating. But such misfires are the exception.
I can envision at least two ways Fieguth’s book could be used in a classroom setting. On its own, it would be a fine primary text for an interdisciplinary course at the intermediate undergraduate level. For a more specialized course tailored to advanced physics majors, the book would naturally complement a more mathematical text in dynamical systems, chaos, or network science. For those texts I can recommend Steven Strogatz’s Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2014), Chaotic Dynamics: An Introduction Based on Classical Mechanics by Tamás Tél and Márton Gruiz (2006), and Albert-László Barabási’s Network Science (2016). But regardless of how Fieguth’s text is used, students of all stripes will find at the end of each chapter a wealth of appropriately challenging exercises ranging from the conceptual to the analytical and computational.
An Introduction to Complex Systems largely accomplishes what it sets out to do. Its application-forward approach is likely to appeal to readers in fields like environmental science and economics, in which complex systems are typically underemphasized but no less important. And if even a specialist like me can read about climate change, lake eutrophication, or any of the panoply of other case studies in this book and exclaim “I never thought about it that way!” then the book should be regarded as a success.
Sean P. Cornelius is a postdoctoral researcher at Northeastern University's Center for Complex Network Research. A physicist by training, he specializes in nonlinear dynamics and networked systems. His research interests include network control, models of emergent behavior, and applications to ecology, infrastructure, and biology.
