Information Technology Reference
In-Depth Information
3
Mostly linear dynamics
One strategy for predicting an improbable but potentially catastrophic future event is
to construct a faithful model of the dynamics of the complex web of interest and study
its extremal properties. But we must also keep in mind that the improbable is not the
same as the impossible. The improbable is something that we know can happen, but
our experience tells us that it probably will not happen, because it has not happened in
the past. Most of what we consider to be common sense or probable is based on what
has happened either to us in the past or to the people we know. In this and subsequent
chapters we explore the probability of such events directly, but to set the stage for that
discussion we examine some of the ways webs become dynamically complex, leading
to an increase in the likelihood of the occurrence of improbable events. The extremes of
a process determine the improbable and consequently it is at these extremes that failure
occurs. Knowing a web's dynamical behavior can help us learn the possible ways in
which it can fail and how long the recovery time from such failure may be. It will also
help us answer such questions as the following. How much time does the web spend
in regions where the likelihood of failure is high? Are the extreme values of dynamical
variables really not important or do they actually dominate the asymptotic behavior of
a complex web?
A web described by linear deterministic dynamics is believed to be completely pre-
dictable, even when a resonance between the internal dynamics and an external driving
force occurs. In a resonance a fictitious web can have an arbitrarily large response to
the excitation and we discuss how such singular behavior is mitigated in the real world.
The idea is that, given complete information about a linear web, we can predict, with
absolute certainty, how that information determines the future. However, we never have
complete information, especially when the number of variables becomes very large, so
we also examine how uncertainty emerges, spreads and eventually dominates what we
can know about the future.
On the other hand, even with complete information a nonlinear web is only partially
predictable, since nonlinear dynamical equations typically have chaotic solutions. The
formal definition of chaos is a sensitive dependence of the solution to the equation
of motion on initial conditions. What this means is that an arbitrarily small change
in how a web is prepared results in a very different final state. The predictability of
chaotic outcomes and the control of phenomena manifesting chaos are examined in the
following chapter with a view towards at least anticipating the unpredictable, such as
the occurrence of the next heart attack; see for example [
45
]. Chaos is mentioned in
