Reflecting recent progress in the study of complex networks, the book under review features modern mathematical developments in the field and numerous real-world applications. In Chapter 1, the authors take the reader on the trip “from the world of Gauss to the world of Pareto” arguing that normal statistics do not describe complex webs. The reason is that phenomena described by normal statistics can be discussed in terms of averages, whereas complex connectivity of webs requires better tools for correct description. Traditional approaches explain uncertainty by means of the maximum-entropy argument where the best interpretation of data is provided by the average which is consistent with the maximum entropy. Replacement of the normal distribution with the hyperbolic distribution emphasizes the key role of the extremes of the data which dominate complex webs. Chapter 2 serves as an introduction to the ubiquity of the inverse power law distribution in the physical, social and life sciences. Nearly fifty complex webs having hyperbolic distributions arising in various disciplines including anthropology, botany, geophysics, psychology and sociology are listed. The properties of data are discussed here along with the methods for data analysis and description. In particular, difference between the observed variation of the data and that which results in normal statistics is considered. Chapter 3 begins with the description of the dynamics of webs, from simple linear dynamical webs to webs of increased complexity. The linear response theory, frequently used in statistical physics for the description of perturbative dynamics, is introduced in this chapter. In addition to stochastic dynamical equations, Fokker-Planck equations are studied. Finally, the authors discuss the replacement of continuous trajectories of Hamiltonian dynamics with the dynamics of discrete events described with the help of the renewal theory and formalism developed in operations research. In Chapter 4, the “briefest of histories of random walks” is used to introduce the concept of a random web. The authors deal with the concepts that were found useful for the description of complex webs like the Poisson statistics, random walks or fractional calculus; the interplay between the randomness and chaos plays an important role here. Dynamics of complex webs in terms of non-analytic functions is the subject of Chapter 5. It is shown that the fractional calculus is vital for the description of the underlying dynamics of fractal processes described by non-analytic functions. Random fields and the fractional propagation of influence on random fields are also studied here. The principal goal of Chapter 6 is “to show how some of the major research items fit into the flow of the general arguments associated with the empirical hyperbolic distributions and their mathematical descriptions in the first five chapters.” Relationship between the small-world theory and hyperbolic behavior of complex webs is discussed along with the idea of a deterministic web in which the complex topology gives rise to the observed scaling behavior. In Chapter 7, the authors determine what is cause and what is effect in many distributions and models discussed in the text. It has been demonstrated that the topology can be a consequence of the underlying dynamical interaction within the network and thus it is not the topology that is causal but the dynamics of the web. The decision-making model that uses the master equation to combine uncertainty by using two-state probabilities for the individual elements and dynamics through the nonlinear time-dependent coupling of these probabilities in time is introduced. This model is employed to explain a number of general principles associated with complex webs, including the Onsager principle and the linear response theory of non-equilibrium statistical physics. The final Chapter 8 provides an overview of the book. Each chapter concludes with an extensive list of references, a very detailed index can be found at the end of the book. The book is well-written, the exposition is crisp and clear. It contains a wealth of fruitful ideas and facts that should boost further studies of complex webs, the topic that has been rapidly developing during the last few decades. Theoretical knowledge is generously illustrated with a variety of applications in engineering, physics, social and natural sciences. This text is a very welcome contribution to the field; it will definitely attract the interest of graduate students, engineers and researchers working in related areas.
Reviewer: Svitlana P. Rogovchenko (Umeå)