Complex Webs: Anticipating the Improbable, Bruce J. West and Paolo Grigolini Cambridge U. Press, New York, 2011. $75.00 (375 pp.). ISBN 978-0-521-11366-3
From taking an airplane to how much we earn, many aspects of our daily lives are connected to webs and networks. That idea is stressed by Bruce West and Paolo Grigolini in their eminently readable inquiry, Complex Webs: Anticipating the Improbable, in which they note the global pursuit by scientists and engineers to develop the field of network science. Past attempts have met with limited and often disappointing results; those attempts include generalized systems theory, complexity theory, catastrophe theory, and the theory of complex adaptive systems. The present search for a network science differs from past efforts in that the theory is now guided by large empirical data sets.
Recent books highlighting different aspects of network science can be roughly separated into popular works that lay out an integrated scientific view of humans and modern technology; manuals and references focused on specific application areas such as biophysics, econophysics, psychophysics, or sociophysics; and texts that explore advanced networks-related topics that go beyond particular disciplines. Complex Webs most closely matches the last (and smallest) category, as it interweaves various topics from statistical physics to support the understanding of complex networks; perhaps in the future those topics will form the foundation of a network science.
Complex Webs is mathematically rigorous, data rich, and entertaining. The first two chapters emphasize that hundreds of complex phenomena dominating our lives have statistical properties described by inverse power laws instead of by the normal Gauss distribution. An unpredictable bridge collapse, the bursting of an economic bubble, or the onset of a heart attack—each is part of a different elaborate web. Gaussian statistics cannot predict those phenomena because such events have their roots in the complexity of webs that represent the flow of such commodities as information, finance, food, and transportation.
This web complexity is manifest in time series that have divergent second moments and that are nonstationary, nonergodic, and non-Poisson. How the new perspective influences fields of investigation such as physiology and bioengineering is an interesting story and provides a context for the authors to introduce many of the mathematical ideas used in understanding webs. For example, in their discussion of fractal physiology and like phenomena, the authors introduce fractal geometry and fractal statistics that follow from the scaling behaviors of power laws.
Our ability to predict the operation of inanimate objects but not of living things means that we can understand the devices cluttering up our world but not much about their relationship to us. To model the lack of understanding, in chapter three the authors provide the physicist’s rationale for randomness. They discuss the shift in prediction from a single trajectory that solves the equations of motion to an ensemble of such trajectories, whose distribution solves the phase-space equations for the probability density. Thus, a preliminary understanding of nature is expressed in terms of averages, fluctuations, and non-Gaussian distributions.
In chapter four, the authors introduce the mathematical techniques used to describe randomness and chaos, and in chapter five they transition to applications of the fractional calculus. There are two distinct strategies for modeling the dynamics of complex webs: “dynamic” dynamics describing the phenomenon and the evolution of the associated probability density. The authors systematically develop both methods and explain the extension into the fractional calculus to incorporate memory; those methods give rise to fractional stochastic differential equations in the first approach and fractional diffusion equations in the second.
Chapter six contains an all-too-brief review of some contemporary developments in network theory. It reflects the authors’ tastes instead of presenting exhaustive coverage of a vast amount of high-quality research from the past decade. The authors tell the familiar story of the progression in our understanding from random networks, to small-world networks, to scale-free networks, and they discuss various measures that allow comparisons of real-world networks to mathematical idealizations.
In chapter seven, the authors discuss recent research findings, including some they themselves have published; they somehow avoid being dogmatic in their presentation. Among the new results described is a generalization of linear response theory clarifying a misconception that appeared in the physics literature a few years ago. The generalization is used to explain how information transfer between complex networks depends on the measures of complexity. It is also used to derive a generalized Onsager principle.
West and Grigolini artfully develop mathematical models for understanding data sets drawn from a variety of venues, and they highlight examples, anecdotes, and historical vignettes that bring the mathematics and its application to life. Consequently, Complex Webs presents a distinctive perspective that makes it stand out. I strongly recommend this remarkable book to those interested in learning the mathematical underpinnings of the science of networks and, more importantly, to those thinking of teaching a course in it.