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small length of the world has the ensuing effect of making this consensus spread
throughout the globe. The small length of random networks does not serve any pur-
pose, insofar as in this case there is no information on the decision-making issue to
transmit from the local to the global level.
What is the role of the scale-free condition in this context? In the literature we have
not found any discussion of this specific issue. In the absence of results of research work
aiming at addressing this specific issue, we have to limit ourselves to making a further
conjecture. We conjecture that the scale-free behavior favors the transmission of local
consensus to the global level, turning the local consensus into a global consensus. This
assumption seems to be supported by the diagram of Figure 6.24 , which shows that the
increase of N yields no significant increase of L and that for
α<
2 the world becomes
impressively small.
However, this hypothesis is not unproblematic. We have seen, in fact, that the deter-
ministic models are hierarchal. A hierarchal web is vulnerable, and it is plausible to
accept the Singer view [ 31 ] that the brain is not hierarchal. Therefore, we have to refine
our search for models that may ensure the fast attainment of global consensus through a
non-hierarchal structure, without ruling out the possibility that the global consensus is
essentially determined by the cooperative action of local communities and that the trans-
formation of local into global consensus may in principle occur without triggering the
scale-free distribution of nodal connections. It would be highly desirable, though, to find
a scale-free model compatible with the laws of Zipf and Pareto, and with a high cluster-
ing coefficient that does not rest on any hierarchal prescription. There are indications in
the study of Internet dynamics that this ideal condition may be possible [ 10 ].
6.5
Problems
6.1 Power laws
Discuss at least four empirical webs that are observed to have an inverse power-law
structure in the context of the model presented in Section 6.1.1 . Identify the empirical
mechanisms that satisfy the properties assumed to be true in the derivation of the model.
6.2 The first-passage time
In Section 6.1.1 we introduced the Kolmogorov backward equation to calculate the
first-passage-time distribution density for a diffusive process. Fill in the details of the
analysis resulting in the biased Gaussian distribution.
6.3 Fractal curves
Redo the calculation depicted in Figure 6.22 . However, instead of inserting two sides
of length 1/3 in the middle third section of the line, replace the middle third section
with three lines of length 1/3 that form a top hat. Iterate this process and note where the
discussion differs from the one given in the text. What is the fractal dimension of the
asymptotic curve?
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