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Reviewer: H. Van Dyke Parunak

Graphs are a seductively simple model for a wide range of natural and artificial systems, including social networks, metabolic and genetic pathways, the Internet, citations in research papers, and transportation systems. The field originated with Euler’s study on the seven bridges of Königsberg in 1736, and for 200 years concentrated on characteristics of specific graphs, often as abstract mathematical objects. The work of Erdös and Rényi on random graphs marked an important transition in the field, shifting the focus from individual structures to the processes that generate them and how graph characteristics depend on those processes. Random processes are conveniently neutral, but evidence has accumulated that many real-world graphs, such as citation networks and patterns of social connections, differ strikingly from random graphs. The 1998 publication of a paper by Watts and Strogatz identified a distinct category of graphs that resemble several natural graphs in having short path lengths (like random graphs), but high node clustering (unlike random graphs). They called these structures small-world networks. The terminological shift from “graph” (a mathematical object) to “network” (an empirical structure) highlights the growing emphasis on using graphs to gain insight about the real world, although many beautiful and abstract mathematical results continue to emerge. The explosion of network studies following Watts and Strogatz was already apparent four years ago (see CR review 128319). Three of the leading researchers in the field have now produced a collection of some of the most important original publications in network science, beginning with the work of Erdös and Rényi, as well as seminal empirical studies from the 1960s. The papers are grouped into four sections. In addition to an overview chapter and a conclusion, the editors have supplied an extensive introduction to each section. The introductory chapter defines the study of networks (as dynamical objects rooted in empirical systems) in relation to the overall history of graph theory, and surveys the rest of the book. Chapter 2, “Historical Developments,” offers a selection of papers from graph theory and social science that anticipate the development of the field of network studies. These older papers identify the giants on whose shoulders current network researchers stand. Chapter 3, “Empirical Studies,” presents exemplary studies of systems as diverse as the World Wide Web (WWW), metabolic networks, scientific collaboration, and sexual networks that reflect the current emphasis on using graphs to understand real-world systems. Chapter 4, “Models of Networks,” provides the fundamental papers on the three main models that have dominated network research: modern derivatives of the random graph model, the small-world model, and scale-free networks, in which the distribution of node degree follows a power law. Chapter 5, “Applications,” documents the practical implications of network science in three areas: understanding and controlling epidemics (both biological and cultural), robustness of network infrastructure (such as communications or transportation) to disruption, and efficiency of searching a network (such as the WWW). The concluding chapter highlights some additional topics that go beyond the papers duplicated in the book, including community structure within networks, hierarchical network structure, assortative mixing, and distinctive features of social networks. The introductions to these sections can be read independently, as a succinct summary of the field. They occupy more than one-sixth of the book’s pages, and cite over 300 references through 2004, including the 44 papers that are reprinted in the volume. A comprehensive index guides readers not only into the introductory sections, but also into the individual papers, resulting in a much more integrated presentation than most reprint collections. This collection will be invaluable, not only in the classroom, but also as a convenient reference for the working practitioner of network science. Online Computing Reviews Service

Reviewer: Harvey Cohn

Network theory is one of the most useful yet evolving branches of applied mathematics (and computer science). This book is essentially a collection of 44 papers that should serve as a seminar text. Even more important, the authors' accompanying exposition synthesizes the subject even for the nonspecialist. As such, this work belongs on every scientific bookshelf. The authors see network theory as having origins in speculative social thought, combined with the concept of a graph-a collection of vertices with special interconnections of lines. The graph was soon regarded as a network, because a dynamic element is introduced by the distance seen in terms of linkages. The basic theme is the small world in a large network. The earliest such theme is the "six degrees of separation" between all humans. This was advanced in 1929 by a Hungarian popular writer Frigyes Karinthy in a bit of whimsy, appropriately called "Chain-Links." (It is reproduced here since no other translation is available.) The authors explain (in US political terms) that all Americans are separated from their congressmen (whom they generally do not know) by three degrees, yet this group, of say 500, facilitates the linkage of hundreds of millions. A more recent small world was produced by the Erdos number, which at first was the length of a collaborative link to the mathematician Paul Erdos, who had an extraordinary number of collaborators (about 500, who in turn had other collaborators, and so on). Even if we bring in the myriad of scientists, their Erdos number is not likely to exceed seven (when it exists). The data are analyzed in one paper to show how the Nobelists facilitate the linkage (although none of them is an Erdos collaborator). Most papers describe modeling experiments that are too numerous to summarize, including, of course, the World Wide Web and reference networks. Also, the study of contagion and immunity creates its own theory. Of these papers, I'll mention only two of general interest: the earliest application paper cited here (1951) is by the physicist Ray Solomonoff and the biologist Anatol Rapoport, "Connectivity of random nets." They consider the fragmentation of a random network into individual components and find that there is a phase transition like crystalization when the mean number of lines at each vertex exceeds one. An early paper showing intensive mathematical rigor is by Paul Erdos and Alfred Renyi (1960), "On the evolution of random graphs." This is on the probable lengths of links in random graphs. Online Computing Reviews Service