Matthew Jackson set an arduous task for himself: to synthesize social scientific research based on the idea of a network. The relevant literature is sprawling. The bibliography has 675 references, the majority from the last 10 years. The author states that “the formal modeling of networks has now reached a maturity across fields…,” yet the focus of each field is different. This makes it difficult to write a balanced and cohesive text. The author has nonetheless maintained cohesion, and the book is very good. Both students and researchers can learn from it.
The book can be divided into three subjects, which cut unevenly across the book’s four sections. The first is graph theory, leading to a discussion of ideas like connectedness, clustering and component size, primarily in the context of random graphs. The second extensively-treated subject is game theory. When compared to traditional game theory, modeling with networks offers a way of describing more intricate payoff functions, and perhaps a more concrete way of modeling. Graph theory is not used much in the game theory examples. In some of these examples, graphs are not involved at all. (There is, however, an extensive discussion of how information spreads through a network, which involves both game and graph theory.) The third, and in my opinion the most interesting subject, is the question of measurement and statistical analysis of real networks. This is the shortest part of the book, discussed primarily in Chapters 3 and 13, although the author is concerned with empirical work throughout.
Jackson is fair-minded, carefully weighing the strengths and the shortcomings of the models, and pointing out important directions for further research. For example, the author makes some incisive criticisms of several algorithms used to reconstruct community structures from data. The principal criticism is that the algorithms create a graph of a social situation without ever specifying what an edge means.
Chapter 3 discusses empirical studies of social networks, and some of the patterns observed. The patterns are intriguing, leading one to ask questions such as “How does the romantic network in a high school compare to the network of co-authors on mathematics papers?” The idea is to identify features that somehow typify social networks in particular applications or in general. Random graph theory is introduced for purposes of comparison, and indeed the social networks do not seem to be formed as random graphs are. It less clear that the features observed, and the measures used to characterize them, can distinguish between different social contexts. This is primarily because the measures that are well-understood are fairly crude. They include the distribution of degrees (of vertices) in the network, and various indices of clustering and connectivity. Further, it is not clear which among several indices is appropriate in a given application. The author has a strong grasp of the difficulties. For example, he applies different measures of clustering to Florentine marriage patterns in the 1400s to highlight different aspects of the problem. There is nothing resembling a consensus as to how networks should be measured. On the other hand, there are many examples in the book that link models, experiments, and social data, and it would seem further progress can be made.
Chapter 13 walks through various algorithms and statistical procedures used to study networks, and the pitfalls faced when using them. Although the statistical analysis of graphs has been intensive in some areas, such as the analysis of evolutionary trees, the state of the art is less developed for social applications. Some of the issues that come up when building statistical models seem to be quite similar to those that come up when one is trying to fit an agent-based simulation to a real-world problem, and there may be some cross-fertilization that can occur here. In any case, there are many needs and open questions that call for further statistical research.
Although the book touches on many areas, at its core it is an economics book. The most extensive section is on game theory. Models of job searches and market structures get detailed treatment. A mathematician may notice the book superficially resembles a mathematics textbook. I am certain it was written in LaTeX, and it often adopts a definition-example-theorem-proof organization. Nonetheless the style is based on a non-mathematical standard. For example, the author gives proofs only when they do not run much past a page. He does give heuristic arguments for all results; sometimes the heuristic explanation is longer than the subsequent proof. One can learn quite a bit from these heuristic arguments, which have a mathematical feel to them. Given that the book is intended for social scientists in general, deemphasizing proofs seems fair. The exposition is wordier than in mathematics texts, but the denser parts can be skimmed without much loss if you have the idea in hand. (Your intrepid reviewer has, of course, read it all.)
The book could be used as a textbook for upper-level mathematics undergraduates, or for first-year graduate students in the social sciences. It is unlikely that either group would have all of the requisite background, but the book is accessible and does not require a thorough understanding of the mathematical underpinnings. To get a sense of what is involved, consider that the author has included appendices that introduce graph theory, infinite series, laws of large numbers, eigenvalues and eigenvectors, Markov chains, generating functions, stochastic dominance, and other topics, all in about 20 pages. Some background in game theory would also be useful.
The exercises are interesting, but perhaps not easy to teach with. I would want to supplement the exercises if I taught from this book, especially if taught to upper-level undergraduates. For one thing, some of the early chapters could use a larger number of exercises, although a supplementary graph theory book could fill this gap. More importantly, some exercises are straightforward if one is properly prepared, but could be difficult if taken on cold. For example, showing that a tree has a unique path between any two given vertices is not hard, but one needs the right idea to start with. Finding the unique subgame perfect equilibrium of the ultimatum game is easy if you have seen some examples of this type of equilibrium (which are not provided in the book), but the answer may seem just wrong to the uninitiated.
If you find the subject matter of the book interesting, I would encourage you to obtain a copy. It is well worth the effort.
John Curran is Assistant Professor of Mathematics at Eastern Michigan University, where he coordinates the actuarial science program. Curran worked for a Wall Street firm for several years before obtaining his Ph.D. in applied mathematics from Brown University.