Don't let people using math-speak bully you into abandoning your common sense. If they cann't explain in simple terms, it's hogwash and they are probably selling snake oil.

While doing some research for a paper elaborating on the theory of ecological and genetic invasions across space, I stumbled on this rather disturbing video about Levy flights.

If you've never heard of a Levy flight, don't worry. Levy flights are like regular random walks but the step lengths are drawn from a power-law-like distribution with infinite variance, such as a Cauchy distribution. Levy flights are special because they not converge to the diffusion equation behavior revealed by Bachelier, Einstein, and Kolmogorov -- instead, they spread faster than a diffusion model predicts. Mandelbrot was initially the main proponent of Levy walks, which have close connections to geometric theory of fractals.

The Center for Homeland Defense and Security is
affiliated with the Naval Postgraduate School (NPS) in Monterey,
California (although the exact financial nature of this affiliation is
unclear to me). What disturbs me about the Center for Homeland Defence
and Security video is it's abuse of obtuse mathematical ideas with claims
of providing simple answers to complex problems like *where will the next
terrorism event occur*. You can develop better models based on common
sense. True Levy flights require people to be able to travel infinitely
fast, which of course can not happen. There is a classic fallacy in
asymptotic analysis which seems to trip people up every generation.

Asymptotic analysis is a mathematical technique that attempts to extract relatively simple common components from a system and use these components to approximate the behavior of the original class of systems. Asymptotic analysis is an incredibly useful and common tool. Calculus is an example of asymptotic analysis where we the concept of infinitessimally small measures to obtain formulas for areas and volumes. The Gauss's normal distribution is an example of asymptotic analysis of sums of random variables. Maximum likelihood analysis is based on asymptotics. The renormalization theory used in physics to study phase transitions is another asymptotic analysis. And so on. Basically, asymptotic results give us a way of filtering the "wheat from the chaff", allowing us to through away complicated parts of an equation that only have a small effect on an answer and keeping a simple formula that gives us a mostly-correct answer about specific questions.

The common fallacy of asymptotic analysis to which I refer is one which
confounds cause and effect. In asymptotic analysis, there are
**many different models** that lead to the same asymptotic formula's. For
instance, there are many different kinds of random walks -- some with
fixed step sizes, some with variable step sizes, some that turn
periodically, some that turn randomly and continuously, ... and as long
as the step sizes are bounded, all of these random walks can be
approximated by the same linear advection-diffusion partial-differential
equations. But what this means for those of us interested in mechanism is
that we can not tell what the underlying mechanism is, based on an
observed match to an asymptotic law -- many different mechanisms would
give the same result.

However, people, even professional researchers and scientists, often mistakenly assume that if the mechanism they have hypothesized gives the right asymptotic statistics, it must be correct. Not so. This would be akin to assuming correlation implied causation. Classical examples of this mistake would be the aristotelian conclusion that the sun revolves around the earth just because of the daily arc it follows across the sky. Nicole Oresme's 14th century criteque laid the logical fallacies of Aristotle's arguments bare, and the Copernican view eventually one the day. Similarly, natural philosophers could not help but conclude that since complicated things must be created, only powerful dieties can create very complicated things, and life itself is a complicate thing, then life must be the creation of a diety. Only when philosophers like Darwin and Wallace proposed another mechanism by which complicated organisms might evolve from simplier ones did the modern science of biology ignite.

The interprettation of power-law statistics as consequences of Levy flights is another (admittedly much smaller, but still problematic) error allong these lines.

This is an issue that really deserves a longer post, as there is a bit of pseudo-science culture around fractal-things that needs some pruning, but the day's work calls. Basic conclusion -- don't let people using math-speak bully you into abandoning your common sense. If they cann't explain it, is probably hogwash.