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arXiv:physics/0606007v3 [physics.soc-ph] 6 Mar 2007

On the Frequency of Severe Terrorist Events

Aaron Clauset

Santa Fe Institute, Santa Fe, NM, USA and

University of New Mexico, Albuquerque, NM, USA.

Maxwell Young

University of New Mexico, Albuquerque, NM, USA.

Kristian Skrede Gleditsch

University of Essex, Wivenhoe Park, Colchester, UK and

Centre for the Study of Civil War, Oslo, Norway.

Summary. In the spirit of Richardson’soriginal (1948) study of the statistics of deadly conflicts,

we study the frequency and severity of terrorist attacks worldwide since 1968. We show that

these events are uniformly characterized by the phenomenon of scale invariance, i.e., the

frequency scales as an inverse power of the severity, P(x) ∝ x−α. We find that this property

is a robust feature of terrorism, persisting when we control for economic development of the

target country, the type of weapon used, and even for short time-scales. Further, we show

that the center of the distribution oscillates slightly with a period of roughly τ ≈ 13 years,

that there exist significant temporal correlations in the frequency of severe events, and that

current models of event incidence cannot account for these variations or the scale invariance

property of global terrorism. Finally, we describe a simple toy model for the generation of these

statistics, and briefly discuss its implications.

Keywords: terrorism; severe attacks; frequency statistics; scale invariance; Richardson’s Law

1. Introduction

Richardson first introduced the concept of scale invariance, i.e., a power-law scaling

between dependent and independent variables, to the study of conflict by examining

the frequency of large and small conflicts, as a function of their severity (Richardson,

1948). His work demonstrated that for both wars and small-scale homicides, the

frequency of an event scales as an inverse power of the event’s severity (in this case,

the number of casualties). Richardson, and subsequent researchers such as Ceder-

man (2003), have found that the frequency of wars of a size x scales as P(x) ∝ x−α,

where α ≈ 2 and is called the scaling exponent. Recently, similar power-law statis-

tics have been found to characterize a wide variety of natural phenomena including

The journal version of this pre-print appeared as “On the Frequency of Severe Terror-

ist Events,” Journal of Conflict Resolution, 51(1):

http://jcr.sagepub.com/cgi/content/abstract/51/1/58.

58 – 88 (2007), which can be found at

Address for correspondence: Aaron Clauset, 1399 Hyde Park Rd., Santa Fe NM, 87501 USA.

E-mail: aaronc@santafe.edu, young@cs.unm.edu, ksg@essex.ac.uk

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disasters such as earthquakes, floods and forest fires (Bak and Tang, 1989; Malamud

et al., 1998; Newman, 2005), social behavior or organization such the distribution of

city sizes, the number of citations for scientific papers, the number of participants in

strikes, and the frequency of words in language (Zipf, 1949; Simon, 1955; Newman,

2005; Biggs, 2005), among others. As a reflection of their apparent ubiquity, but

somewhat pejoratively, it has even been said that such power-law statistics seem

“more normal than normal” (Li et al., 2006).

In this paper, we extend Richardson’s program of study to the most topical kind

of conflict: terrorism. Specifically, we empirically study the distributional nature

of the frequency and severity of terrorist events worldwide since 1968. Although

terrorism as a political tool has a long history (Congleton, 2002; Enders and Sandler,

2006), it is only in the modern era that small groups of so-motivated individuals

have had access to extremely destructive weapons (Shubik, 1997; Federal Bureau of

Investigation, 1999). Access to such weapons has resulted in severe terrorist events

such as the 7 August 1998 car bombing in Nairobi, Kenya which injured or killed

over 5200, and the more well known attack on 11 September 2001 in New York City

which killed 2749. Conventional wisdom holds that these rare-but-severe events are

outliers, i.e., they are qualitatively different from the more common terrorist attacks

that kill or injure only a few people. Although that impression may be true from

an operational standpoint, it is false from a statistical standpoint. The frequency-

severity statistics of terrorist events are scale invariant and, consequently, there is

no fundamental difference between small and large events; both are consistent with

a single underlying distribution. This fact indicates that there is no reason to expect

that “major” or more severe terrorist attacks should require qualitatively different

explanations than less salient forms of terrorism.

The results of our study are significant for several reasons. First, severe events

have a well documented disproportional effect on the targeted society. Terrorists

typically seek publicity, and the media tend to devote significantly more attention

to dramatic events that cause a large number of casualties and directly affect the

target audience (Wilkinson, 1997; Gartner, 2004). When governments are uncertain

about the strength of their opponents, more severe terrorist attacks can help terror-

ist groups signal greater resources and resolve and thereby influence a government’s

response to their actions (Overgaard, 1994). Research on the consequences of ter-

rorism, such as its economic impact, likewise tends to find that more severe events

exert a much greater impact than less severe incidents (Enders and Sandler, 2006,

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On the Frequency of Severe Terrorist Events3

Ch. 9). For instance, Navarro and Spencer (2001) report dramatic declines in share

prices on the New York Stock Exchange, Nasdaq, and Amex after the devastating

11 September attacks in the United States. In contrast, although financial markets

fell immediately following the 7 July 2005 bombings in London, share prices quickly

recovered the next day as it became clear that the bombings had not been as se-

vere as many initially had feared.1Recent examples of this non-linear relationship

abound, although the tremendous reorganization of the national security appara-

tus in the United States following the 11 September 2001 attacks is perhaps the

most notable in Western society. Second, although researchers have made efforts

to develop models that predict the incidence of terrorist attacks, without also pre-

dicting the severity, these predictions provide an insufficient guide for policy, risk

analysis, and recovery management. In the absence of an accurate understanding

of the severity statistics of terrorism, a short-sighted but rational policy would be

to assume that every attack will be severe. Later, we will show that when we adapt

current models of terrorism to predict event severity, they misleadingly predict a

thin tailed distribution, which would cause us to dramatically underestimate the

future casualties and consequences of terrorist attacks. Clearly, we need to better

understand how our models can be adapted to more accurately produce the ob-

served patterns in the frequency-severity statistics. That is, an adequate model of

terrorism should not only give us indications of where or when events are likely to

occur, but also tell us how severe they are likely to be. Toward this end, we describe

a toy model that can at least produce the correct severity distribution.

Past research on conflict has tended to focus on large-scale events like wars, and

to characterize them dichotomously according to their incidence or absence, rather

than according to their scale or severity. This tendency was recently highlighted

by Cederman (2003) for modeling wars and state formation, and by Lacina (2006) for

civil wars. Additionally accounting for an event’s severity can provide significantly

greater guidance to policy makers; for instance, Cioffi-Revilla (1991) accurately

predicted the magnitude (the base ten logarithm of total combatant fatalities) of

the Persian Gulf War in 1991, which could have helped in estimating the political

consequences of the war.

As mentioned above, research on terrorism has also tended to focus on inci-

dence, rather than severity. Recently, however, two of the authors of this study

1See figures for the FTSE 100 index of the 100 largest companies listed on the London Stock

Exchange at http://www.econstats.com/eqty/eq d mi 5.htm.

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demonstrated for the first time that the relationship between the frequency and

severity of terrorist events exhibits the surprising and robust feature of scale in-

variance (Clauset and Young, 2005), just as Richardson showed for wars. In a

subsequent study, Johnson et al. (2005) considered data for fatal attacks or clashes

in the guerilla conflicts of Colombia and Iraq, suggesting that these too exhibit

scale invariance. Additionally, they claim that the time-varying behavior of these

two distributions are trending toward a common power law with parameter α = 2.5

– a value they note as being similar to the one reported by Clauset and Young

(2005) for terrorist events in economically underdeveloped nations. Johnson et al.

then adapted a dynamic equilibrium model of herding behavior on the stock market

to explain the patterns they observed for these guerilla conflicts. From this model,

they conjecture that the conflicts of Iraq, Colombia, Afghanistan, Casamance (Sene-

gal), Indonesia, Israel, Northern Ireland and global terrorism are all converging to

a universal distribution with exactly this value of α (Johnson et al., 2006). We will

briefly revisit this idea in a later section. Finally, the recent work of Bogen and

Jones (2006) also considers the severity of terrorist attacks primarily via aggregate

figures to assess whether there has been an increase in the severity of terrorism over

time, and to forecast mortality due to terrorism.

This articles makes three main contributions. First, we make explicit the util-

ity of using a power-law model of the severity statistics of terrorist attacks, and

demonstrate the robust empirical fact that these frequency-severity statistics are

scale invariant. Second, we demonstrate that distributional analyses of terrorism

data can shed considerable light on the subject by revealing new relationships and

patterns. And third, we show that, when adapted to predict event severity, existing

models of terrorism incidence fail to produce the observed heavy-tail in the severity

statistics of terrorism, and that new models are needed in order to connect our

existing knowledge about what factors promote or discourage terrorism with our

new results on the severity statistics.

2. Power laws: a brief primer

Before plunging into our analysis, and for the benefit of readers who may be un-

familiar with the topic, we will briefly consider the topics of heavy-tailed statistics

and power-law distributions. What distinguishes a power-law distributions from

the more familiar normal distribution is its heavy tail, i.e., in a power law, there

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is a non-trivial amount of weight far from the distribution’s center. This feature,

in turn, implies that events orders of magnitude larger (or smaller) than the mean

are relatively common. The latter point is particularly true when compared to a

normal distribution, where essentially no weight is far from the mean. Although

there are many distributions that exhibit heavy tails, the power law is a particularly

special case, being identifiable by a straight line with slope α on doubly-logarithmic

axes2, and which appears widely in physics. The power law has the particular form

in which multiplication of the argument, e.g., by a factor of 2, results in a propor-

tional division of the frequency, e.g., by a factor of 4, and the ratio of these values

is given by the “scaling parameter” alpha. Because this relationship holds for all

values of the power law, the distribution is said to “scale”, which implies that there

is no qualitative difference between large and small events.

Power-law distributed quantities are actually quite common, although we often

do not think of them as being that way. Consider, for instance, the populations

of the 600 largest cities in the United States (from the 2000 Census). With the

average population being only ?x? = 165 719, metropolises like New York City

and Los Angles would seem to be clear “outliers” relative to this value. The first

clue that this distribution is poorly explained by a truncated normal distribution

is that the sample standard deviation σ = 410 730 is significantly larger than

the sample mean. Indeed, if we model the data in this way, we would expect to

see 1.8 times fewer cities at least as large as Albuquerque, at 448 607, than we

actually do. Further, because it is more than a dozen standard deviations from

the mean, we would never expect to see a city as large as New York City, with a

population of 8 008 278; for a sample this size, the largest city we would expect

to see is Indianapolis, at 781 870. Figure 1 shows the actual distribution, plotted

on doubly-logarithmic axes, as its complementary cumulative distribution function

(ccdf) P(X ≥ x), which is the standard way of visualizing this kind of data.3The

scaling behavior of this distribution is quite clear, and a power-law model (black

line) of its shape is in strong agreement with the data. In contrast, the truncated

normal model is a terrible fit.

2A straight line on doubly-logarithmic axes is a necessary, but not sufficient condition for a

distribution to be a power law; for example, when we have only a small number of observations

from an exponentially distributed variable, it can appear roughly straight on double-logarithmic

axes.

3The ccdf is preferable to the probability distribution function (p.d.f) as the latter is signifi-

cantly noisier in the upper tail, exactly where subtle variations in behavior can be concealed. If a

distribution scales, it will continue to do so on the ccdf