From Osher Doctorow

Aaron Clauset and Maxwell Young of U. New Mexico Dept. of Computer
Science in "Scale invariance in global terrorism," physics/0502014 v2 1
May 2005 present an enormous amount of data derived from the MIPT
database which in turn comes from the Rand Terrorism Chronology and
Incident Database (respectively 1968-1997, 1998-present), and which
indicate that the so-called scale invariant equation:

1) P(x) = (or of the order of) x^(-2)

characterizes terrorism between 1968 and 2004. The equation is
technically scale invariant in a fractal sense, which I'll try to
discuss later, and Clauset and Young take the model somewhat further by
considering mixtures with slightly different power laws like (1)
depending on the type of weapon. Here X is the severity of attacks by
injuries and deaths, and P(X = x) is taken to be P(x), that is to say
the probability that X is greater than or equal to each value x.

Physicists will undoubtedly be most interested in the causation behind
these models, and among them are (from p. 4 of their paper):

A. Rare Events or Large Deviations or Heavy/Fat Tails
B. Scaling or scaling invariance, closed related to self-similarity
(see Wolfram's Mathworld under Self-Similarity) which yields a power
law like y = x^a with a the Hausdorff dimension.
C. Competition between states and non-state actors (non-state
terrorists)
D. Type of weapon and industrialization versus non-industrialization to
account for the exponent, where explosives seem to be the key
demarcating aspect in type of weapon as distinct from fire, firearms,
knives, chemical/biological weapons, other.

There is considerable emphasis on A as well as the other points, and
criticism of the previous models which regarded Rare Events as merely
"outliers" rather than important for themselves.

An interesting derivation of the power law (1) which casts considerable
light on the causes is done on page 4 of their paper, and roughly
speaking equates p(x)dx = p(s)ds where x refers to actual events and s
to potential events (and p(x) potential severity distribution). The
"reasonable" assumption is made that p(s) has an exponential
distribution exp(as) with a 0 up to some maximum s and that the
likelihood of an event being successful is inversely related to its
potential severity so that:

2) x = exp(bs), b 0

in which case we get:

3) p(x) = x^(-alpha), alpha = 1 - a/b

and when /a/ = /b/, we get:

4) p(x) = x^(-2)

Why does p(x)dx = p(s)ds? According to the authors, one can assume
that some but not all events are actually executed, the proportion due
to such things as collective counter-terrorism actions by states,
random failures, social factors, etc., so that p(x)dx = p(s)ds or
something like it is indicated.

Osher Doctorow