Introduction to the Modeling and Analysis of Complex Systems
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About the Textbook
Introduction to the Modeling and Analysis of Complex Systems introduces students to mathematical/computational modeling and analysis developed in the emerging interdis- ciplinary field of Complex Systems Science. Complex systems are systems made of a large number of microscopic components interacting with each other in nontrivial ways. Many real-world systems can be understood as complex systems, where critically impor- tant information resides in the relationships between the parts and not necessarily within the parts themselves.
This textbook offers an accessible yet technically-oriented introduction to the modeling and analysis of complex systems. The topics covered include: fundamentals of modeling, basics of dynamical systems, discrete-time models, continuous-time models, bifurcations, chaos, cellular automata, continuous field models, static networks, dynamic networks, and agent-based models. Most of these topics are discussed in two chapters, one focusing on computational modeling and the other on mathematical analysis. This unique approach provides a comprehensive view of related concepts and techniques, and allows readers and instructors to flexibly choose relevant materials based on their objectives and needs. Python sample codes are provided for each modeling example.
About the Author
, D.Sc., is an Associate Professor in the Department of Systems Science and Industrial Engineering, and the Director of the Center for Collective Dynamics of Complex Systems (CoCo), at Binghamton University, State University of . He received his BSc, MSc and DSc in Information Science, all from the University of Tokyo, Japan. He did his postdoctoral work at the Complex Systems Institute in Cambridge, Massachusetts, from 1999 to 2002. His research interests include complex dynamical networks, human and social dynamics, collective behaviors, artificial life/chem- istry, and interactive systems, among others.
He is an expert of mathematical/computational modeling and analysis of various com- plex systems. He has published more than 100 peer-reviewed journal articles and confer- ence proceedings papers and has edited eight books and conference proceedings about complex systems related topics. His publications have acquired more than 2000 citations as of July 2015. He currently serves as an elected Board Member of the International Society for Artificial Life (ISAL) and as an editorial board member for Complex Adaptive Systems Modeling (SpringerOpen), International Journal of Parallel, Emergent and Dis- tributed Systems (Taylor & Francis), and Applied Network Science (SpringerOpen).
Reviewer’s Notes
This book provides an excellent introduction to the field of modeling and analysis of com- plex systems to both undergraduate and graduate students in the physical sciences, social sciences, health sciences, humanities, and engineering. Knowledge of basic mathemat- ics is presumed of the reader who is given glimpses into the vast, diverse and rich world of nonlinear algebraic and differential equations that model various real-world phenomena. The treatment of the field is thorough and comprehensive, and the book is written in a very lucid and student-oriented fashion. A distinguishing feature of the book, which uses the freely available software Python, is numerous examples and hands-on exercises on complex system modeling, with the student being encouraged to develop and test his or her own code in order to gain vital experience.
The book is divided into three parts. Part I provides a basic introduction to the art and science of model building and gives a brief historical overview of complex system modeling. Part II is concerned with systems having a small number of variables. After introducing the reader to the important concept of phase space of a dynamical system, it covers the modeling and analysis of both discrete- and continuous-time systems in a systematic fashion. A very interesting feature of this part is the analysis of the behavior of such a system around its equilibrium state, small perturbations around which can lead to bifurcations and chaos. Part III covers the simulation of systems with a large number of variables. After introducing the reader to the interactive simulation tool PyCX, it presents the modeling and analysis of complex systems (e.g., waves in excitable media, spread of epidemics and forest fires) with cellular automata. It next discusses the modeling and analysis of continuous fields that are represented by partial differential equations. Exam- ples are diffusion-reaction systems which can exhibit spontaneous self-organizing behav- ior (e.g., Turing pattern formation, Belousov-Zhabotinsky reaction and Gray-Scott pattern formation). Part III concludes with the modeling and analysis of dynamical networks and agent-based models.
The concepts of emergence and self-organization constitute the underlying thread that weaves the various chapters of the book together.
About the Reviewer: Dr. . Chatterjee received his Bachelor’s Degree in Tech- nology (Honors) from the Indian Institute of Technology, Kharagpur, India, and M.S. and Ph.D. degrees from Rensselaer Polytechnic Institute, Troy, , USA, all in Chem- ical Engineering. He has taught a variety of engineering and mathematical courses and his research interests are the areas of philosophy of science, mathematical modeling and simulation. Presently he is Associate Professor in the Department of Paper and Biopro- cess Engineering at SUNY College of Environmental Science and Forestry, Syracuse,
. He is also a Fellow of the Institution of Engineers (India) and Member of the Indian Institute of Chemical Engineers.
Sayama has produced a very comprehensive introduction and overview of complexity. Typically, these topics would occur in many different courses, as a side note or possible behavior of a particular type of mathematical model, but only after overcoming a huge hurdle of technical detail. Thus, initially, I saw this book as a “mile-wide, inch-deep” ap- proach to teaching dynamical systems, cellular automata, networks, and the like. Then I realized that while students will learn a great deal about these topics, the real focus is learning about complexity and its hallmarks through particular mathematical models in which it occurs. In that respect, the book is remarkably deep and excellent at illustrating how complexity occurs in so many different contexts that it is worth studying in its own right. In other words, Sayama sort of rotates the axes from “calculus”, “linear algebra”, and so forth, so that the axes are “self-organization”, “emergence”, etc. This means that I would be equally happy to use the modeling chapters in a 100-level introduction to mod- eling course or to use the analysis chapters in an upper-level, calculus-based modeling course. The Python programming used throughout provides a nice introduction to simula- tion and gives readers an excellent sandbox in which to explore the topic. The exercises provide an excellent starting point to help readers ask and answer interesting questions about the models and about the underlying situations being modeled. The logical struc- ture of the material takes maximum advantage of early material to support analysis and understanding of more difficult models. The organization also means that students expe- riencing such material early in their academic careers will naturally have a framework for later studies that delve more deeply into the analysis and application of particular mathe- matical tools, like PDEs or networks.
About the Reviewer: Dr. earned his Ph.D. in applied mathematics from the Uni- versity of Arizona. Since then, he has earned the rank of full professor at St. College where he often teaches differential equations, mathematical modeling, multivari- able calculus and numerical analysis, as well as a variety of other courses. He has guided a number of successful undergraduate research projects related to modeling of complex systems, and is currently interested in applications of such models to education, both in terms of teaching and learning and of the educational system as a whole. Outside of the office, he can often be found training in various martial arts or enjoying life with his wife and two cats.
This is an introductory textbook about the concepts and techniques of mathematical/com- putational modeling and analysis developed in the emerging interdisciplinary field of com- plex systems science. Complex systems can be informally defined as networks of many interacting components that may arise and evolve through self-organization. Many real- world systems can be modeled and understood as complex systems, such as political organizations, human cultures/languages, national and international economies, stock markets, the Internet, social networks, the global climate, food webs, brains, physiolog- ical systems, and even gene regulatory networks within a single cell; essentially, they are everywhere. In all of these systems, a massive amount of microscopic components are interacting with each other in nontrivial ways, where important information resides in the relationships between the parts and not necessarily within the parts themselves. It is therefore imperative to model and analyze how such interactions form and operate in order to understand what will emerge at a macroscopic scale in the system.
Complex systems science has gained an increasing amount of attention from both in- side and outside of academia over the last few decades. There are many excellent books already published, which can introduce you to the big ideas and key take-home messages about complex systems. In the meantime, one persistent challenge I have been having in teaching complex systems over the last several years is the apparent lack of accessible, easy-to-follow, introductory-level technical textbooks. What I mean by technical textbooks are the ones that get down to the “wet and dirty” details of how to build mathematical or computational models of complex systems and how to simulate and analyze them. Other books that go into such levels of detail are typically written for advanced students who are already doing some kind of research in physics, mathematics, or computer science. What I needed, instead, was a technical textbook that would be more appropriate for a broader audience—college freshmen and sophomores in any science, technology, engineering, and mathematics (STEM) areas, undergraduate/graduate students in other majors, such as the social sciences, management/organizational sciences, health sciences and the hu- manities, and even advanced high school students looking for research projects who are
interested in complex systems modeling.
This OpenSUNY textbook is my humble attempt to meet this need. As someone who
didn’t major in either physics or mathematics, and who moved away from the mainstream of computer science, I thought I could be a good “translator” of technical material for laypeople who don’t major in those quantitative fields. To make the material as tangible as possible, I included a lot of step-by-step instructions on how to develop models (espe- cially computer simulation codes), as well as many visuals, in this book. Those detailed instructions/visuals may sometimes look a bit redundant, but hopefully they will make the technical material more accessible to many of you. I also hope that this book can serve as a good introduction and reference for graduate students and researchers who are new to the field of complex systems.
In this textbook, we will use Python for computational modeling and simulation. Python is a rapidly growing computer programming language widely used for scientific computing and also for system development in the information technology industries. It is freely available and quite easy to learn for non-computer science majors. I hope that using Python as a modeling and simulation tool will help you gain some real marketable skills, and it will thus be much more beneficial than using other pre-made modeling/simulation software. All the Python sample codes for modeling examples are available from the textbook’s website at http://bingweb.binghamton.edu/~sayama/textbook/, which are directly linked from each code example shown in this textbook (if you are reading this electronically). Solutions for the exercises are also available from this website.
To maintain a good balance between accessibility and technical depth/rigor, I have written most of the topics in two chapters, one focusing on hands-on modeling work and the other focusing on more advanced mathematical analysis. Here is a more specific breakdown:
Preliminary chapters 1, 2
Modeling chapters 3, 4, 6, 10, 11, 13, 15, 16, 19 Analysis chapters 5, 7, 8, 9, 12, 14, 17, 18
The preliminary and modeling chapters are marked with an orange side bar at the top, while the analysis chapters are marked with a blue side bar at the bottom. The modeling chapters won’t require any in-depth mathematical skills; some basic knowledge of deriva- tives and probabilities is enough. The analysis chapters are based on a more solid under- standing of calculus, differential equations, linear algebra, and probability and statistics. I hope this unique way of organizing topics in two complementary chapters will provide
a comprehensive view of the related concepts and techniques, as well as allow you to flexibly choose relevant materials based on your learning/teaching objectives and needs.
If you are an instructor, here are some suggested uses for this textbook: • One-semester course as an introduction to complex systems modeling
– Target audience: College freshmen or sophomores (or also for research projects by advanced high school students)
– Chapters to be covered: Part I and some modeling chapters selected from Parts II & III
• One-semester course as an introduction to dynamical systems
– Target audience: Senior undergraduate or graduate students
– Chapters to be covered: Parts I & II, plus Continuous Field Models chapters (both modeling and analysis)
• One-semester advanced course on complex systems modeling and analysis
– Target audience: Graduate students who already know dynamical systems
– Chapters to be covered: Part III (both modeling and analysis)
• Two-semester course sequence on both modeling and analysis of complex systems
– Target audience: Senior undergraduate or graduate students – Chapters to be covered: Whole textbook
Note that the chapters of this textbook are organized purely based on their content. They are not designed to be convenient curricular modules that can be learned or taught in similar amounts of time. Some chapters (especially the preliminary ones) are very short and easy, while others (especially the analysis ones) are extensive and challenging. If you are teaching a course using this book, it is recommended to allocate time and resources to each chapter according to its length and difficulty level.
One more thing I need to note is that the contents of this book are focused on dynam- ical models, and as such, they are not intended to cover all aspects of complex systems science. There are many important topics that are not included in the text because of the limitation of space and time. For example, topics that involve probability, stochasticity, and statistics, such as information theory, entropy, complexity measurement, stochastic models, statistical analysis, and machine learning, are not discussed much in this book. Fortunately, there are several excellent textbooks on these topics already available.
This textbook was made possible, thanks to the help and support of a number of people. I would like to first express my sincere gratitude to Leod, the former Chair of the Department of Bioengineering at Binghamton University, who encouraged me to write this textbook. The initial brainstorming discussions I had with him helped me tremendously in shaping the basic topics and structure of this book. After all, it was Ken who hired me at Binghamton, so I owe him a lot anyway. Thank you Ken.
My thanks also go to Yaneer Bar-Yam, the President of the Complex Systems Institute (NECSI), where I did my postdoctoral studies (alas, way more than a decade ago—time flies). I was professionally introduced to the vast field of complex sys- tems by him, and the various research projects I worked on under his guidance helped me learn many of the materials discussed in this book. He also gave me the opportunity to teach complex systems modeling at the NECSI Summer/Winter Schools. This ongoing teaching experience has helped me a lot in the development of the instructional materials included in this textbook. I would also like to thank my former PhD advisor, – anagi, former Professor at the University of Tokyo. His ways of valuing both scientific rigor and intellectual freedom and creativity influenced me greatly, which are still flowing in my blood.
This textbook uses PyCX, a simple Python-based complex systems simulation frame- work. Its GUI was developed by , a former undergraduate student at Bingham- ton University and now an MBA student at the University of Maryland, and Przemysław Szufel and Bogumił Kamin ́ski, professors at the Warsaw School of Economics, to whom I owe greatly. If you find PyCX’s GUI useful, you should be grateful to them, not me. Please send them a thank you note.
I thank , , , and all others who have made this wonderful OpenSUNY textbook program possible. Having this book with open access to everyone in the world is one of the main reasons why I decided to write it in the first place. Moreover, I greatly appreciate the two external reviewers, , at St. College, and . Chatterjee, at SUNY College of Environmental Science and Forestry, whose detailed feedback was essential in improving the quality and accu- racy of the contents of this book. In particular, ’s very thorough, constructive, extremely helpful comments have helped bring the scientific contents of this textbook up to a whole new level. I truly appreciate his valuable feedback. I also thank for her very careful copy editing for the final version of the manuscript, which greatly im- proved the quality of the text.
My thanks are also due to my fellow faculty members and graduate students at the Center for Collective
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