Biomedical Engineering Reference
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was entirely in terms of discrete time series, the theory can be extended straight-
forwardly to spatially extended dynamical systems (196), where it quantifies
self-organization (197), to controlled dynamical systems and transducers, and to
prediction problems generally (188).
8.4. Power Law Distributions
Over the last decade or so, it has become reasonably common to see people
(especially physicists) claiming that some system or other is complex, because it
exhibits a power law distribution of event sizes. Despite its popularity, this is
simply a fallacy. No one has demonstrated any relation between power laws and
any kind of formal complexity measure. Nor is there any link tying power laws
to our intuitive idea of complex systems as ones with strongly interdependent
parts.
It is true that, in equilibrium statistical mechanics , one does not find power
laws except near phase transitions (200), when the system is complex by our
standard. This has encouraged physicists to equate power laws as such with
complexity. Despite this, it has been known for half a century (5) that there are
many, many ways of generating power laws, just as there are many mechanisms
that can produce Poisson distributions, or Gaussians. Perhaps the simplest one
is that recently demonstrated by Reed and Hughes (201), namely exponen-
tial growth coupled with random observation times. The observation of power
laws alone thus says nothing about complexity (except in thermodynamic equi-
librium!), and certainly is not a reliable sign of some specific favored mecha-
nism, such as self-organized criticality (202,203) or highly optimized tolerance
(204-206).
8.4.1.
Statistical Issues Relating to Power Laws
The statistics of power laws are poorly understood within the field of com-
plex systems, to a degree that is quite surprising considering how much attention
has been paid to them. To be quite honest, there is little reason to think that
many of the things claimed to be power laws actually are such, as opposed to
some other kind of heavy-tailed distribution. This brief section will attempt to
inoculate the reader against some common mistakes, most of which are related
to the fact that a power law makes a straight line on a log-log plot. Since it
would be impractical to cite all papers that commit these mistakes, and unfair to
cite only some of them, I will omit references here; interested readers will be
able to assemble collections of their own very rapidly.
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