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consensus and that the scale-free distribution density of links is related to the transmis-
sion of information from one cluster to another so as to transform local consensus into
global synchrony.
In the literature of complex networks there is a widely shared conviction that webs
with large clustering coefficients that are scale-free are generated for the main purpose
of reaching the highest possible degree of efficiency. In this section we are interested in
determining whether the concept of topological efficiency corresponds to the perhaps
more intuitive notion of dynamical efficiency, interpreted as the condition favoring the
realization of global consensus. From an intuitive point of view, a large clustering coef-
ficient corresponds to the existence of local tight-knit communities that are expected to
generate local consensus. However, for the consensus to become global, it is necessary
to have links connecting different clusters. This raises the question of whether or not the
scale-free nature of real webs is generated by the communication between clusters. If
this is the case, however, we have to establish whether it can be realized without recourse
to a hierarchical prescription. We impose this constraint because according to Singer
[
31
] the brain is an orchestra without a conductor, suggesting that hierarchical networks
are not a proper model to describe the most complex of networks, the human brain.
6.2.1
Random webs
Random-web models are similar in spirit to random-walk models in that their mathe-
matical analysis does not capture the full richness of reality; they are not intended to,
but they provide a solvable mathematical setting in which to answer specific questions.
Consequently, completely random webs provide a useful starting point for investigating
how elements arbitrarily distributed in space influence one another. This model is not
the real world, but becoming facile with it can provide a tool with which to better under-
stand the real world. Where real-world data yield parameters that deviate markedly from
the model, the model mechanism on which that parameter is based can be discarded or
modified. Measures of such things as clustering, betweenness and web size all serve to
distinguish real-world from random webs.
Random webs are an example of totally disordered systems. According to a widely
accepted perspective, random webs are at one end of a continuum with completely
ordered webs at the other, locating complex webs somewhere in between. Consequently
a totally random web is not complex. However, we have to establish full command of
their properties, so as to properly address issues such as those discussed in the previous
section.
Some definitions
Random networks were widely studied by Erdös and Rényi in 1959, 1960 and 1961. For
an accurate reference to the work of these two outstanding mathematicians we refer the
readers to the review paper [
2
]. In this subsection we develop some elementary notions
that will make life easier as we proceed. A graph is a pair of sets
G
≡{
P
,
E
}
;
P
is a set
of vertices (points, nodes)
P
1
,
P
N
and
E
is a set of edges (links or lines) that
connect two elements of
P
. Figure
6.6
shows a graph with five points and two edges.
P
2
,...,
