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put forward by West and Goldberger [
37
]. In addition there are tracts on information net-
works, exploring the interface between man and machine as did Wiener, who initiated
the modern theory of communication through his 1948 topic
Cybernetics
[
39
].
Each of the above webs and others are generated by mechanisms specific to the com-
plex phenomenon being considered, the psychology of human interaction in the social
domain; the connectivity across scales in the natural sciences, such as across neurons
in the brain; and the explosive growth of information in nonlinear dynamical networks
through the generation of chaos. On the other hand, there are properties common to all
these webs that enable us to identify them as such. These common features, if suffi-
ciently robust, may form the foundation of a science of networks, to revert back to the
accepted vocabulary. In this topic we focus on the fact that to understand how these
complex webs change over time the traditional analysis of information flow through
a web, involving the exponential distributions of message statistics and consequently
Poisson statistics of traffic volume, must be abandoned and replaced with less familiar
statistical techniques. The techniques we introduce and discuss have appeared in the
diaspore of the physical-sciences literature on networks and complexity and we shall
call on them when needed.
Some excellent review articles on complex webs have been published recently, each
with its own philosophical slant, and each of which is different from the perspective
developed herein. Albert and Barabási [
1
] review complex webs from the perspective
of non-equilibrium statistical physics, giving a rigorous foundation for the small-world
models discussed in the lay literature [
4
,
5
,
34
,
35
]. In some ways this approach is a
natural extension of Lotka's approach, but it concentrates on what has been learned
in the intervening century. Newman [
24
] organized his review around the various
applications that have been made over the past decade, including social, information,
technology and biology webs and the mathematical rendition of the common properties
that are observed among them. The more recent survey by Costa
et al
.[
7
] views com-
plex network research as being at the intersection of graph theory and statistical physics
and focuses on existing measurements that support the principal models found in the
literature.
The approach taken herein is a hybrid of the above views and the ideas were originally
put into perspective by us with the able assistance of the then student Elvis Geneston
and may be found in condensed form in West
et al.
[
38
]. We took non-equilibrium sta-
tistical physics and its techniques as our starting point, used recent experiments done on
complex webs to highlight the inadequacies of the traditional approaches, and applied
these data to motivate the development of new mathematical modeling techniques nec-
essary to understand the underlying web structure. Much of that work has been folded
into the present pages, but including all the background, gaps and details not contained
in the original journal publication. This was done having the student in mind.
For clarity it is useful to list the properties associated with complex webs, because
we are seeking a quantitative measure that may include an ordinal relation for web
complexity. We note, however, that in everyday usage phenomena with complicated
and intricate features having both the characteristics of randomness and order are called
complex. Furthermore, there is no consensus on what constitutes a good quantitative
