The most recommended books about mathematical models

Peter Turchin has gotten famous recently for predicting the US political upheaval of 2020 way back in 2012.

This book represents the first landmark of Turchin’s attempt to understand the ebbs and flows of history using dynamical models. The book’s centerpiece is a formalization of a theory about how empires rise and fall, first conceived by the 14th century (!) Arab scholar Ibn Khaldun.

The book inspired me to replicate the computational model it presents, and it was remarkably illuminating to watch empires grow, fight, and collapse on my computer screen. 

One of the earliest books on this subject, Uri Alon presents an engaging account of biological networks. Focussing on transcriptional networks, and their motifs, the book illustrates the nexus between network structures and functions. The second edition of the book launched a few years ago and has some updated content and new material on interesting functionalities such as fold change detection. Uri Alon is a very accomplished scientist, mentor, and a leader in the field of biological networks/systems biology.

As a math student I found economics a slippery subject and, therefore, was hesitant to read any book on the subject. 

Theory of Value is a short, formal manuscript, that includes the definition of an economy. It was the first book I read in economics and I loved it. It induced me to move to New York and to study the field.

We increasingly live in a virtual world of models – pandemic models, climate change models, economic models. These have taken on what seems like a greater reality than the world of people and places we know and what used to be called common sense. 

At one level everyone feels this. Modeling is Thompson’s job. This leaves her ideally placed to grapple with the question of just how real these models are and what weight we should put on these alternate realities and what weight on a reality we grew up in, which feels like its disappearing.

Thompson tackles these complex topics in a down-to-earth way. The title Escape from Model Land tells you she’s on our side rather than the side of the experts.

I am an economist by training. I taught economic courses at Rutgers University for two decades but the only economic framework I consider useful for real-world managers is the classical school known as Austrian Economics, of which Murray Rothbard was one of the more eloquent and accessible voices. Mainstream economists busy themselves with complex mathematical models that are in the end utterly useless in explaining the world, or worse, are utterly wrong in predicting the consequences of policies but are provided by “court economists” to support ideologies. Murry Rothbard, in contrast, lays down the principles of economic theory that will guide your investment/competitive strategies based on the most solid and realistic economic reasoning around. The book is not easy to read, but it will change your understanding of the world. It inspired me to get a Ph.D. 

I appreciated how this book took on the challenge of applying advanced mathematical modeling and simulation techniques to gain new insights into the social evolution of norms and institutions in societies, which is a critical topic in many fields, including conflict and peace economics.

I learned much from the book’s coverage of selected conflict and peace topics such as riots, revolutions, the 2010 Arab Spring movement, and the social evolution of cooperation. I especially like how the mathematical models in the book are described intuitively using wonderfully imaginative diagrams.

In this way, both the mathematically and non-mathematically inclined can come away with a richer understanding of human behavior in dynamically changing social systems.

I always find myself applying algorithmic thinking in my everyday life—it affects the way I put away dishes, navigate to the store, and organize my to-do lists. And I think others could benefit from that mindset.

So, when I read this book, my reaction was "Yes! That's what I want people to know. I just wish I could have said it that well!" The authors (who I know, but didn't know they wrote a book together), did a fantastic job of selecting algorithms with deep human connections. Really! And they explain them just right, without getting too mathematical but while still hitting the key ideas with clarity and accuracy. Fantastic!

Now we are getting to the modern era, where physics meets computing in ways never dreamed of by the founders of quantum physics. Seth Lloyd is a professor at MIT and one of the first to prove that quantum theory admits the possibility of building computers on wholly quantum foundations. He is also one of the most creative and engaging writers on the subject. His main message is that information is the basis of all things physical. From quantum bits or qubits to cosmology, this book has it all, and is written in a highly accessible way. Not a single equation can be found. 

What do pandemics, climate change, extreme weather, financial crises, wealth inequality, and social media all have in common? They are all well described by heavy-tail statistics, which you may have never heard about and were almost certainly never taught in your introductory statistics class. The Fundamentals of Heavy Tails is the first text that attempts to close this gap in undergraduate STEM education. This well-written text is a wonderful blend of intuition and rigorous results. The reader will be pleasantly surprised to learn that heavy-tail distributions are neither rare nor mysterious and are a natural result of multiplicative random processes.

Finally, for those who are not afraid of math, they should read this book because there is a lot of heavy-duty math. The good news for the rest of us is you can ignore all the math and still get a lot out of it, especially knowledge about market microstructure and how to find the theoretically optimal trading strategies given some assumptions about the price dynamics. Even if you don’t want to or can’t solve those darn stochastic differential equations, you can still implement a numerical approximation. At the minimum, you will learn common trading lingo such as “walking the book” or “the ITCH feed”.