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this chapter on linearity so the reader does not fall into the trap of thinking that random
fluctuations are always the result of what we do not know or cannot measure. Random
fluctuations can also result from nonlinear dynamics.
We start the discussion of dynamics with the simplest web and incrementally increase
its complexity until we exhaust the modes of analysis available for understanding its
linear dynamics. It is not our intention to provide a complete discussion of all these
techniques, but rather to indicate how particular kinds of complexity defy traditional
methods of analysis and therefore require new techniques for their understanding. We
did some of that in the first two chapters by introducing non-analytic functions and
pointing out that because they were non-differentiable they could not be solutions to dif-
ferential equations of motion. In this chapter, among other things, we suggest one way
to study the dynamics of fractals using the fractional calculus, but postpone discussion
of that calculus until Chapter 5.
In this chapter we examine ways of incrementally increasing the complexity of a
web, beginning with the linear dynamics of mechanical networks that can be solved
exactly. Such finite-dimensional webs can be put in contact with infinite-dimensional
environments, whose influence on the web is to introduce a random force. In this way
the web is described by a set of stochastic differential equations. This approach was used
by Langevin at the beginning of the last century to describe the physical phenomenon
of diffusion in terms of the forces moving the diffusing particles around. Of particular
interest is the long-time influence of the correlations in the random force on the web
dynamics. We do not restrict our focus to physical webs, but instead use diffusion as a
generic concept to examine the behavior of aggregates of people, bugs and information
packets as well as particles. In this way we examine how a web's dynamics responds to
both external and internal potentials.
Two approaches have been developed to describe the dynamics of complex webs in
physics, namely the one mentioned above, which involves the equations of motion for
the dynamical web variables, and a second technique that examines the dynamics of the
corresponding probability density. The equations for the dynamical variables trace out
the behavior of a single trajectory of the complex web, whereas the probability density
determines how an ensemble of such trajectories evolves over time in phase space. The
equations for the ensemble distribution function in physics are called Fokker-Planck
equations (FPEs) and are essentially equivalent to the Langevin equation in describ-
ing the web dynamics. Here again we discuss how the FPEs can be used outside the
physics domain to study such things as the social phenomenon of decision-making. We
discuss in some detail a model of choosing between two alternatives using a process of
aggregating uncertain information until one is sufficiently certain to make a decision.
Another way to increase the complexity of the web is through the statistical proper-
ties of the random force. With this in mind we start from the simple Poisson process,
which we subsequently learn describes the statistics of random networks, and generalize
the statistics to include the influence of memory. As an example of the importance of
memory we examine the phenomenon of failure through the use of time-dependent fail-
ure rates, the application of inverse power-law survival probabilities and autocorrelation
functions.
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