Before I enter into a more detailed review, let me briefly explain, why I found the book
“Network Science” by AlbertLászló Barabási particularly intriguing for a review.
Over the last twenty years the topic of networks has picked up considerable interest in various sciences. Barabási is undoubtedly a pioneer in this regard. He
has fostered new developments since the surge of network science has began. He is also a coauthor
of one of the most cited reviews (with Reka Albert) [1]. This review,
in tandem with the review by Newman [2], defined the groundwork for a decade of networks research. Some other pioneers, such as Newman [3],
Kleinberg [4] and Vespignani [5], have written
or collaborated on textbooks but there is definitely still a need to have additional sources.
In this regard, one should also mention that Barabási has been supported by an entire
research team in the preparation of this book and I shall mention Márton Pósfai explicitly
here, who is credited in the book with the responsibility for “calculations, simulations and
measurements”.
So what is the target, and the target audience, of Barabási’s most recent take on the
subject? The first important point to note is that “Network Science” is first and foremost
a textbook. From the introduction it is evident that the goal of classroom use was
a design principle. More precisely, a course taught by Barabási for upperlevel
undergraduates and beginning graduates in physics, computer science and engineering forms
the main thread of the book. The book includes very nice homework exercises
as well as a detailed strategy for a final student project. The goal of the project is
to determine, sample, and analyze a real network. This type of project already hints at
the core idea to synergize network theory and data analysis. In fact, this link
(no pun intended) between realworld data sets and deceptively simple theoretical models was
a key reason why the modern version of network science has revitalized the existing classical
analysis of random graphs from the 1950s and 1960s. Barabási’s book reflects this development
in an excellent fashion.
The book has ten chapters. Although this grouping might be very simplistic, essentially chapters 29 revolve around the structure of networks. This is highlighted by the
central discussion of the powerlaw of many networks in chapter 4. More precisely, the
probability \(p_k\) to find a node of degree \(k\) in the network is given by \(p_k\sim k^{\gamma}\) for some exponent \(\gamma\). This theme is always compared to classical random graphs,
a la ErdösRényi, which follow a binomial (or limiting Poisson) degree distribution.
Barabási uses the comparison in a wide variety of ways to explain other key discoveries
such as the existence of hubs, pathlengths/smallworld properties, network growth models,
fault and attack tolerance, community structure, and so on. The gradual development and
careful presentation of the network structure theme is very accessible, yet conveys many
deep insights found during the last twenty years. Therefore, anybody interested in learning
this topic from the ground up—with almost no prerequisities required—should definitely
consider reading Barabási’s book, or use it as a course text.
It is also important to mention that the book has many interesting features which distinguish
it from other, more traditional, textbooks. The book does not follow the classical ‘definition,
lemma, proof, theorem, proof, corollary’style common for mathematics texts. However, even from the viewpoint
of physics and engineering, it has many relatively uncommon approaches. The layout mainly proceeds
along four columns. The two inner columns contain the main text. The outer two columns, sometimes
even more, are used for figures, remarks, summary boxes, data sets, visualization, links to
online resources, etc. Also, the personal introduction by Barabási is a very interesting read, and
I won’t say more about it here to not spoil the first encounter with the preface. Another
important point is that the book is available freely online:
barabasi.com/networksciencebook.
Each chapter is followed by ‘Advanced Topics’, which
discuss some of the derivations of mathematical formulas and statements.
This appendixstyle is also used to deepen several discussions, which cannot find their way into
the main text. In some sense, this appendixstyle certainly helps to bridge the backgrounds of
diverse audiences quite a bit, and it is easier to distinguish the main application ideas from
the mathematical theory. This is certainly a possible innovative approach that makes the book
useful for transdisciplinary courses. My personal recommendation is that for a
mathematicallyoriented course, one can bring in most advanced topics or even augment the
arguments by ideas grounded in theoretical probability from the textbook of Durrett [6]
or the recent monograph of van der Hofstad [7].
Since this review has been prepared for the dynamical systems community, let me
comment on aspects related to dynamics. The last part of the book, chapter 10, is
probably closest to dynamics. It discusses epidemic spreading and how the structure of the
network influences the location of the epidemic threshold. In particular, one may make the
epidemic threshold move quite a bit, or even disappear in a limit, when random graphs are replaced
by powerlaw degree distributions. In this context a possible meanfield approximation for
various susceptibleinfected dynamics and its variants is sketched. Of course, we are very
far away from a deep understanding of dynamical processes on and of
networks [8], so it may actually be a wise decision at this stage by Barabási to
not include an indepth look at dynamics in a first course. However, it would certainly be more
than welcome—at least in my personal opinion—that the applied dynamics community should think
about writing one, if not more than one, sequel to the books mentioned in the references as well as
the book by Barabási to present the fundamentals of dynamics for network science.
In summary, I can highly recommend Barabási’s book for anyone looking to teach an interdisciplinary
course on networks or anyone aiming to learn the basics of network structure with a view towards
applications. Personally, I intend to use it as a source for undergraduate seminars for mathematics
students to introduce them to the topic by selfstudy. Hopefully the book is going to set out
the exploration path for them to acquire and invent new mathematical methods. As another personal note,
let me stress that the field is moving so rapidly, that it seems useful to take stock once in
a while. From a mathematical perspective, carefully preparing and
categorizing recent developments in a more accessible form is beneficial
at the time this book review is written. Hence, Barabási sets out with his textbook an agenda
that hopefully many others are going to follow soon.
Bibliography

R. Albert and A.L. Barabási.
Statistical mechanics of complex networks.
Rev. Mod. Phys., 74:4797, 2002.

M.E.J. Newman.
The structure and function of complex networks.
SIAM Review, 45:167256, 2003.

M.E.J. Newman.
Networks  An Introduction.
OUP, 2011.

D. Easley and J. Kleinberg.
Networks, Crowds, and Markets: Reasoning about a highly connected world.
CUP, 2010.

A. Barrat, M. Barthélemy, and A. Vespignani.
Dynamical Processes on Complex Networks.
CUP, 2008.

R. Durrett.
Random Graph Dynamics.
CUP, 2010.

R. van~den Hofstad.
Random Graphs and Complex Networks: Volume 1.
CUP, 2016.

T. Gross and H. Sayama, editors.
Adaptive Networks: Theory, Models and Applications.
Springer, 2009.