Interested in leveraging the popularity of social networking sites like Facebook to spark your liberal arts students’ interest in mathematics? How about piquing the interest of a serious sociology, psychology or biology student with the promise that math can serve as a powerful tool for extracting meaning in their disciplines? Or perhaps you have an eager math graduate student who might benefit from a one-stop shop offering an advanced, up-to-date treatment of a single, timely mathematical subfield? Or maybe you are a mathematician already working in the field of network theory and would appreciate a single, current resource — complete with an up-to-date, 336-source bibliography — of advancements in your field? Well, look no further than M. E. J. Newman’s Networks: An Introduction.
Newman’s 772-page tome succeeds with an ambitious agenda to speak to many different kinds of readers seeking information about networks. In Part I — The Empirical Study of Networks — the reader gains a bird’s eye view of network theory that is accessible even to readers with limited math backgrounds. There is no number crunching, jargon or heavy-duty math notation here; just lots of accessible prose on par with articles written for the Science section of The New York Times. Of course, the absence of calculations means that there are no exercises in Part I. This, however, is more than made up for by the copious number of examples of networks — including technological, social, information and biological networks — that dominate. Numerous color and black-and-white figures assist the reader in acquiring a comprehensive, not to mention enviable, view of the place and usefulness of networking in modern society. The interested reader will find exercises on which to work in every other part of the book.
Part II — Fundamentals of Network Theory — is where the visible math enters. After offering a precise definition of a network — a collection of vertices joined by edges — Newman makes rigorous the mathematics behind networks. Mathematicians have various tools — such as weighted or unweighted adjacency matrices — and various measures — such as degree centrality — for representing and finding patterns in networks. Even as Newman introduces the mathematics behind networks, however, he continues to employ a conversational tone. In addition, he is careful to ground the theory in applications. The figure depicting a friendship network at a US high school by race, for example, offers a vivid and revealing example of the math behind assortive mixing. As the old expression goes, a picture — in this case, a mathematical picture — is worth a thousand words.
In Part III — Computer Algorithms — readers are extended the invitation to get their hands dirty understanding and writing programs that will assist in the management of the often unwieldy data sets involving real-world networks. Of course, Newman is quick to point out the existence of numerous high-quality, prepackaged software programs for managing network data. However, he makes the valid point that researchers need to understand such computer programs and algorithms, lest their research questions be restricted to those that can be answered using only available software packages.
It is with this eye towards research that, in Parts IV — Network Models — and Part V — Processes on Networks — Newman brings the reader up to date with current advances in the field. These two parts would be suitable for graduate students in math or researchers interested in recent advancements in network theory. Newman himself acknowledges that the final chapters “probably raise as many questions as they answer.” Of course, this fact essentially ensures the role that Networks: An Introduction will play in moving the body of knowledge that is network theory forward in the years to come.
Susan D’Agostino is an Assistant Professor of Mathematics at Southern New Hampshire University. She has written articles for The Chronicle of Higher Education, MAA Focus and Math Horizons. She is currently writing a book with mathematical themes intended for an audience of nonmathematicians.