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

From Osher Doctorow

I should emphasize that X and S, or x and s as values of them
respectively, are roughly speaking frequency versus intensity of
terrorist events respectively. The authors point out that there is a
well established relationship between the frequency and intensity of
wars, citing 4 papers on this, including 2 by the famous L. F.
Richardson (1948, 1960), one by J. S. Levy (1983), and one by D. C.
Roberts and D. L. Turcotte (1998).

The authors indicate that small groups of non-state actors may have
access to more destructive potential in industrialized nations than
elsewhere.

Osher Doctorow

From Osher Doctorow

Kenneth Falconer's (U. Bristol, U.K.) Fractal Geometry, Wiley:
Chichester 1990, p. 27, has for Hausdorff measures (which generalize
length, area, volume) H:

1) H^s(kF) = k^a H^s(F)

where kF = {kx: x in F}, th set F scaled by a factor k. He points out
that for ordinary length, area, volume, scaling properties are well
known, e.g., length when magnified by a factor k is multiplied by k for
a curve, area of plane region is multiplied by k^2m volume of
3-dimensional object by k^3, so (1) has s-dimensional Hausdorff measure
scaling with factor k^s. Note that a Holder condition with exponent
a, namely /f(x) - f(y)/ = c/x - y/^a, with constants c and a 0
implies H^(s/a)(f(F)) = c^(s/a)H^s(F) by Proposition 2.2 of Falconer
(p. 27) and scaling properties are fundamental to the theory of
fractals.

Note that if S1 to Sm are contractions, i.e., /Si(x) - Si(y)/ = c/x -
y/ for all x in a closed subset D of R^n, then a subset F of D is
called invariant for the transformations Si if F = U Si(F) where the
union goes from i 1 to m. Such invariant sets are often fractals, and
Falconer shows that families of contractions, also called iterated
function schemes, define unique non-empty compact invariant sets such
as the Cantor middle thirds set. In fact, Falconer describes a general
construction for fractals via the above apparatus and proves its main
theorem.

Osher Doctorow