Information Technology Reference
In-Depth Information
7
Dynamics of chance
In this chapter we explore web dynamics using a master equation in which the rate
of change of probability is determined by the probability flux into and out of the
state of interest. The master equation captures the interactions among large numbers
of elements each with the same internal dynamics; in this case the internal dynam-
ics consist of the switching of a node between two states. A two-state node may be
viewed as the possible choices of an individual, say whether or not that person will
vote for a particular candidate in an election. This is one of the simplest dynamical
webs which has been shown mathematically to result in synchronization under cer-
tain well-defined conditions. We focus on the intermittent fluctuations emerging from
a phase-transition process that achieves synchronized behavior for the strength of the
interaction exceeding a critical value. This model provides a first step towards proving
that these intermittent fluctuations, rather than being a nuisance, are important chan-
nels of information transmission allowing communication within and between different
complex webs. The crucial power-law index
discussed earlier and there inserted for
mathematical convenience is here determined by the web dynamics. This observation
on the inverse power-law index leads us to define the network efficiency in a form
that might not coincide with earlier definitions proposed through the observation of the
network topology.
Both the discrete and the continuous master equation are discussed for modeling web
dynamics. One of the most important aspects of the analysis concerns the perturbation
of one complex network by another and the transfer of information between complex
clusters. A cluster is a network with a uniformity of opinion due to a phase transition
to a given state. This modeling strategy is not new, but in fact dates back to Yule [
71
],
who used the master-equation approach, before the approach had been introduced, to
obtain an inverse power-law distribution. We investigate whether the cluster's opinion
is robust or whether it can be easily changed by perturbing the way members of the
cluster interact with one another.
A general theme of this chapter is to develop the point of view (already briefly intro-
duced in Section 3.2.2) that linear response theory (LRT) suitably generalized can be
considered a universal principle. Specifically, we generalize the LRT of non-equilibrium
statistical physics to faithfully determine the average web response to a perturbation
and we examine the evidence for such a generalization. The latter part of the chapter is
devoted to proving this remarkable theorem. This would be the first such principle that
is apparently independent of the specific type of complex web being considered. This
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