Abstract
The heydays of what is known as mathematical sociology were probably in the 1960s and 1970s, two decades that saw the publication of Coleman’s 1964 Introduction to Mathematical Sociology and Fararo’s 1973 Mathematical Sociology: An Introduction to Fundamentals; two now classic and equally impressive although rather distinctive books. If we did a headcount of the number of self-proclaimed mathematical sociologists nowadays, we would certainly be led to think that mathematical sociology was far off the mainstream. At the same time, mathematical, statistical and computational sophistication play a central role in contemporary state-of-the-art sociology.
Bonacich and Lu’s Introduction to Mathematical Sociology grew out of an undergraduate course taught at UCLA. It is self-contained and introduces the required mathematics along the way. A simple motivation to learn something about the subject paired with a background in basic high school mathematics is enough to guide the reader through the 18 chapters. The reward is a basic but solid insight into mathematical sociology. The price for keeping the text at an elementary level, however, is the omission of calculus. This may be the reason why the authors have emphasized the finite mathematics of social network analysis. This choice makes sense; network analysis is one of the most vibrant fields of research in contemporary sociology. In this respect, the book is closer to Fararo’s than Coleman’s book. The distinction between theory and method is not emphasized, and when they write about network analysis, it leans more towards the latter than the former. The authors still strive to present the reader with some formal language and the building blocks that allow for systematic and precise theorizing about complex social phenomena.
Besides an emphasis on relations and networks, covering equivalence, centrality, balance and cliques (Chapters 4, 5, 6, 9, 10, 13), onto small-world and scale-free networks (Chapters 11, 12), the book also has chapters on demographic models (Chapter 15), game theory and exchange (Chapters 16, 17) and complexity (Chapter 18). Some of the mathematical chapters will be hard to sell to the average sociology student, such as probability theory (Chapter 3), matrix manipulation (Chapter 7) and Markov chains (Chapter 14), but overall the book has a fruitful mix of mathematical introductions and sociological applications.
It does suffer from a few lapses. I am surprised that the authors do not provide a useful bibliography and some good suggestions for further reading. There is no doubt that they could have done this without much effort, and the publisher should have insisted on it. The editing and copyediting is also sloppy: The first-person narrator is sometimes singular sometimes plural, chapter demonstrations are missing online, and there is an annoying number of typos. These mistakes are an embarrassment to Princeton University Press and highly unfair to the authors.
Moreover, and perhaps more importantly, Bonacich and Lu should have worked more on the sociological intuition and the sociological examples. Some of them are far from the interests of most sociologists and some seem headless. For instance, why would a sociologist be interested in analysing the spread of disease or the life-cycle of cats? I can give my students good arguments for taking an interest in the former, but I would find it harder to argue for the latter. Unfortunately, the authors do not make an effort to rescue either one. Introduction to Mathematical Sociology is better when it borders social theory, for example making a connection between small-world networks and Giddens’s theory of structuration. I would have liked to see more such ‘loose’ applications of analytical thinking to mainstream sociological problems. When discussing the Markov assumption for example, the authors could have provided more thoughts on the role of history in sociology, and the degree to which and in what situations it is reasonable to disregard it. They could also have discussed how mathematical tools can handle history, reminding the reader that this book does not tell the whole story about the potential of mathematical sociology.
There are no similar books in print, so it is hard to make comparisons. As opposed to the two classics mentioned earlier, the first thing to note is that the book under review is less ambitious. Secondly, where Coleman strived for a fairly broad coverage, albeit a bit haphazard, Fararo was much more systematic and focused. Bonacich and Lu take the latter approach, added to that a strong drive to make the reading experience interactive. Most chapters end with exercises and simple simulations of various sorts that couple the printed book to online supporting material and the free Mathematica Player. The book does not cover any programming, but the examples still provide a glimpse into the potential utility of computer simulation tools, including the intriguing Schelling model of segregation. The basic mathematical material is of course available in many places, perhaps most notably in a handful of volumes in Sage’s Quantitative applications series (the thin green ones), but Bonacich and Lu’s effort to be accessible do set their book apart from these purely mathematical texts.
Compared to other introductions to social network analysis, the book shines. Introduction to Mathematical Sociology gives a more solid theoretical foundation than most other introductions to social network analysis, which tend to jump directly into the empirical analysis of social network data. The book gives the reader a set of analytical tools for theorizing social structure at the micro and macro levels. That background will undoubtedly put students in a much better position to analyse social networks both theoretically and empirically. Sociology will never morph into a technical discipline such as economics, which is perhaps all for the better. Still, in my opinion sociologists should follow what goes on in the social sciences at large. It would also enhance sociology if we improved our ability to communicate with the more technical and formalized sciences. For those who share these beliefs, Bonacich and Lu’s book is a fine way to get started.