Data Assimilation: A Mathematical Introduction
As an applied mathematician or theoretical physicist it is easy to get excited about networks. Here is a research field that is directly relevant to a wide variety of applications, while real progress can still be made with pen and paper. What is driving this progress is “network magic”, a wide variety of mathematical tricks and clever approximations that can tame the inherent complexity of large networks and distill it down into manageable systems of equations. While rarely mathematically exact, these techniques often provide a close enough approximation to shed light on important phenomena.
All of the above is particularly true in dynamical networks. By combining network structure with dynamics, dynamical networks offer a versatile framework that can faithfully capture the structure and dynamics of real-world systems. The tools of both dynamical systems theory and network science can then be used to run highly efficient simulations or gain analytical insights.
For some time dynamical networks has had a relatively steep learning curve as working in this cross-disciplinary field required understanding concepts from statistical physics, nonlinear dynamics and computer science. However, recently accessible introductions have started to appear. A new (and powerful) addition to this literature is the book “Dynamical Systems on Networks: A Tutorial” by Mason Porter and James Gleeson.
The first thing the reader notices about this book is how short it is. With just 56 pages of main text this is a quick read. However, despite its short length the book is a treasure trove of valuable insights and provides a fairly comprehensive introduction. The authors manage this partly by writing in a clear and concise style. The text goes straight to the point, but still provides enough context to be easily accessible. Particularly readers with experience in nonlinear dynamics will likely find it an easy read, while network scientists might wish to supplement it with an introductory dynamics textbook.
The book starts with an introduction to central concepts and paradigmatic models. Chapter 2 takes the reader straight to the mathematical methods which occupy center stage. The chapter introduces a variety of powerful tools including master stability functions, percolation-based methods, and moment expansions. Later chapters provide concise reviews of software tools and packages, advanced topics and open questions.
What impressed me most about this book is that the authors have made very good choices regarding the level of included detail.
Both authors are experts in the field who have extensive (and to some degree complementary) experience in working with network models and dynamical processes. It is evident that they have a very clear grasp on what information the reader really needs. For example, they include an insightful discussion of synchronous vs. asynchronous updates in simulations. This is in itself perhaps not an exciting topic and is hence glossed over in most research papers, but it is also something that anybody seeking to work in the field needs to understand.
The book can be described as a mathematical text in the sense that theoretical physics papers are mathematical. It contains equations and sophisticated, sometimes rigorous, math but not detailed proofs. For topics such as moment expansions the authors have managed to capture the mathematical ideas without getting lost in mathematical detail.
Some finer points are included in appendices and the book is rounded off by an itemized overview of books and review articles in the field, which many readers will find very valuable.
In summary, the authors have done an excellent job of compiling almost everything a researcher really needs to know to get started in the field in a very short text. I will give this book to every new PhD student starting in my lab and warmly recommend it to everybody starting to work in the beautiful field of network dynamics.