**
PREFACE
**

This book is about the growth and structure of random networks.
The book

is written by physicists and
presents the point of view of a physicist,
but is

addressed to all researchers involved in this subject and students.

Where was physics 50 years ago, and where is it now?
At first sight, the

role of physics is decreasing; other natural sciences are developing more rapidly.

However, physics has penetrated into all sciences. A natural step for a physicist

is to jump from the traditional topics of physics to new intriguing problems.

Actually, our book describes a flight from physics to the new interdisciplinary

field of networks. This escape is, however,
still dependent on physics.

For many years the term `random graphs' usually meant to mathematicians

static, `equilibrium' networks with a Poisson-type distribution of connections.

Mathematicians have made truly great advances in the description of such

networks.

Only recently have we realized that we reside in a world of networks.
The

Internet and World Wide Web (WWW) are changing our lives. Our physical

existence is based on various biological networks. The extent of the development

of communications networks is a good indicator of
the level of development in a

country.
`Network' turns out to be a central notion in our time, and
the explosion

of interest in networks is already
a social and cultural phenomenon.

Graph theory has made great progress.
However, the most important natural

and artificial networks have a
specific architecture based on a fat-tailed

distribution of the number of connections of vertices
that differs crucially from the

`classical random graphs' studied by mathematicians.
As a rule, these networks

are not static but evolving objects. Their state is far from equilibrium and their

structure cannot be understood without insight into the principles of their

evolution.
Only in the last few
years have physicists started extensive empirical and

theoretical research
into
networks organized in such a way.
Earlier, physicists'

interest was rather in neural and Boolean networks
where
the arrangement of

connections was secondary.

We think that the physics approach
is the most advantageous for understanding

the evolution of networks. Actually, what we physicists are now doing on this

active topic is a direct generalization of the usual physics of growth, percolation

phenomena,
diffusion, self-organized criticality, mesoscopic systems, etc.

Our aim is to understand networks: that is, to understand the basic principles

of their structural organization and evolution.
We believe that this understanding

is necessary to find the best solutions to the problems of real networks.

We decided to present a concise informative book which could be used even

by students without a deep knowledge of mathematics and statistical physics

and which would be a good source of reference material.
Therefore we have tried

to introduce the main ideas and concepts in as simple a manner as possible, with

minimal mathematics.
Special attention is given to real networks, both natural

and artificial.
We discuss in detail the collected empirical data and numerous

real applications of existing theories. The urgent problems of communication

networks are highlighted and discussed.

For a description of network evolution, we prefer to use a simpler continuum

approach.
We feel that it is more important to be understood than to
be

perfectly rigorous.
Also, we follow the hierarchy of values in Western science: an

experiment and empirical data are more valuable than an estimate; an estimate

is more valuable than an approximate calculation; an approximate calculation

is more valuable than a rigorous result.
More cumbersome calculations and

supplementary materials are placed in appendices. We hope that all of the results

and statements that we discuss can be easily found in the text and understood

without undertaking detailed calculations.
Therefore, we ask our brave readers

to skim over difficult pages without hesitation and not to pay any attention to

footnotes.
However, as this is a monograph written by theoretical physicists, we

try to keep a `physical level' of strictness in our explanations and definitions.

Although, we try to avoid superfluous words,
we are not afraid to repeat important

statements at a different level.
We hope that the book will also be useful to

mathematicians, as a source
of interesting new objects and ideas.

We thank our friends and colleagues for their help. Foremost among these are

our collaborators in this field: Alexander V. Goltsev and Alexander N. Samukhin

from the Ioffe Institute in St Petersburg.
We did not reprint figures with

empirical data from original papers but made sketches of data. We are grateful to

Albert-L'aszl'o Barab'asi, Stefan Bornholdt, Jonathan Doye, Jennifer Dunne, Lee

Giles, Ramesh Govindan, Byungnam Kahng, Ravi Kumar, Neo Martinez, Sergei

Maslov, Mark Newman, Sidney Redner, Ricard Sol'e, Alessandro Vespignani, and

their coauthors
for permission to use data from their original figures for derivative

reproduction.
We are much indebted to John Bulger, Ester Richards, Chris

Fowler, David Duckitt, Goutam Tripathy, and Neville Hankins, the copy editor

at Oxford University Press for correcting the English of our book.
Our computers

did not crash only thanks to Miguel Dias Costa and Joao Viana Lopes.

When this book was written, one of us (SND) was on leave from his native Ioffe

Institute, and he acknowledges the Centre of Physics of Porto for their support

and hospitality.

PortoWWWWWWWWWWWWWWWWWWWWWWWWWWWWWu. S.N.D.

May 2002WWWWWWWWWWWWWWWWWWWWWWWWWWW J.F.F.M.