Synopsis - Complex Webs: Anticipating the Improbable - page 363

Chapter 6 quantify certain web properties and are a consequence of the web topology,
whether the web is random or small-world, or any of the other choices available,
and consequently the topology is interpreted as causal. We showed that the topology
can be a consequence of the underlying dynamical interaction among the elements
of the network and therefore it is the dynamics of the web which are causal, not the
topology. Of course, we could have parsed the discussion into the multiple causes
identified and discussed by Aristotle, but we did not. The decision-making model
reviewed in Chapter 7 uses the master equation to combine uncertainty through the
use of two-state probabilities for the individual elements and dynamics through the
nonlinear time-dependent coupling of these probabilities in time. This model is shown
to manifest phase transitions in which the apparently free-running behaviors of the
individual nodes become synchronized, switching between the two states collectively.
This consensus of opinion is seen to result in a distribution of state changes that is an
inverse power law for a number of decades in time followed by exponential behavior.
The extent of the inverse power law in time depends on the number of elements in the
web, as does the size of the fluctuations. In fact the fluctuations observed in the web are
produced by the web having a finite number of nodes rather than by thermal excitations
as they are in physical networks.
The decision-making model was used in the demonstrations of a number of general
principles associated with complex webs. One was a generalization of the Onsager
principle in which the traditional exponential relaxation of a spontaneous fluctu-
ation in a physical process is replaced by the inverse power-law relaxation of a
spontaneous fluctuation in complex webs. Another was the generalization of LRT
of non-equilibrium statistical physics, where the historical basis of LRT, namely
continuous dynamical trajectories, was replaced with the discrete dynamics of crit-
ical events and renewal statistics. The familiar phenomenon of stochastic resonance
was shown to be a consequence of this new LRT, as were various microscopic
phenomena [ 2 ]. A final application of LRT was in the explanation of the psychophysical
phenomenon of habituation, through the statistical habituation model (SHM) whereby
a simple stimulus is shown to be attenuated as an inverse power law in time by
a complex web. The form of the stimulus, whether it is heard, tasted, touched or
smelled, determines the ergodic nature of the complex neuronal webs and conse-
quently the form of habituation in the SHM. Finally, we have an explanation as to
why the uninterrupted drone of the lecturer puts even the most dedicated student to
sleep.
References
[1] P. Allegrini, M. Bologna, P. Grigolini and B. J. West, “Fluctuation-dissipation theorem for
event-dominated processes,” Phys. Rev. Lett . 99 , 010603 (2007).
[2] P. Allegrini, M. Bologna, L. Fronzoni, P. Grigolini and L. Silvestri, “Experimental quench-
ing of harmonic stimuli: universality of linear response theory,” Phys. Rev. Lett. 103 ,
030602 (2009).