arXiv:physics/0606007v3 [physics.soc-ph] 6 Mar 2007
On the Frequency of Severe Terrorist Events
Santa Fe Institute, Santa Fe, NM, USA and
University of New Mexico, Albuquerque, NM, USA.
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
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):
58 – 88 (2007), which can be found at
Address for correspondence: Aaron Clauset, 1399 Hyde Park Rd., Santa Fe NM, 87501 USA.
E-mail: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
2 Clauset, Young and Gleditsch
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,
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.
4 Clauset, Young and Gleditsch
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
On the Frequency of Severe Terrorist Events5
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
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
28 Clauset, Young and Gleditsch
transformed data, i.e., one takes the log of both the dependent and independent
variables, or one could bin the data into decades, and then measures the slope using
a least-squares linear fit. Unfortunately, this procedure yields a biased estimate
for the scaling exponent (Goldstein et al., 2004). For continuous power-law data,
Newman (2005) gives an unbiased estimator based on the method of maximum
likelihood; however, it too yields a biased estimate when applied to discrete data
like ours. Goldstein et al. (2004) studied the bias of some estimators for power-law
distributed data, and, also using the method of maximum likelihood, give a tran-
scendental equation whose solution is an unbiased estimator for discrete data. In
our main study, we use a generalization of this equation as our discrete maximum
To give the reader a sense of the performance of these methods, we show in
Figure 6b the results of applying them to simulated data derived from the dis-
crete generator described above. Quite clearly, the discrete maximum likelihood
estimator yields highly accurate results, with the other techniques either over- or
under-estimating the true scaling parameter, sometimes dramatically so. Johnson
et al. (2006) have also studied the accuracy of these estimators, but apparently only
for data derived from the continuous deviate generator described above.
The discrete maximum likelihood estimator of Goldstein et al. assumes that
the tail encompasses the entire distribution. A generalization of their formula to
distributions where the tail begins at some minimum value xmin≥ 1 follows, and
the value of αML that satisfies this equation is the discrete maximum likelihood
where the xiare the data in the tail, n is the number of such observations, and
ζ(α,xmin) is the incomplete Riemann zeta function. If desired, the latter can be
rewritten as ζ(α) − Hα
xmin, being the difference between a zeta function and the
xminth harmonic number of order α. When xmin= 1, the left-hand side reduces to
ζ′(α)/ζ(α), the values of which can be calculated using most standard mathematical
software. Alternatively, one can numerically maximize the log-likelihood function
On the Frequency of Severe Terrorist Events29
L(α | x) = −nlogζ(α,xmin) − α
which may be significantly more convenient than dealing with the derivative of the
incomplete zeta function. This approach is what was used in both the present study,
and in our preliminary study of this terrorism data (Clauset and Young, 2005).
These equations assume that the range of the scaling behavior, i.e., the lower
bound xmin, is known. In real-world situations, this value is often estimated visually
and a conservative estimate of such can be sufficient when the data span a half-dozen
or so orders of magnitude. However, the data for many social or complex systems
only span a few orders of magnitude at most, and an underpopulated tail would
provide our tools with little statistical power. Thus, we use a numerical method for
selecting the xminthat yields the best power-law model for the data. Specifically, for
each xminover some reasonable range, we first estimate the scaling parameter αML
over the data x ≥ xmin, and then compute the Kolmogorov-Smirnov (KS) goodness-
of-fit statistic between the data being fit and a theoretical power-law distribution
with parameters αMLand xmin. We then select the xminthat yields the best such
fit to our data. For simulated data with similar characteristics to the MIPT data,
we find that this method correctly estimates both the lower bound on the scaling
and the scaling exponent. Mathematically, we take
where F(x;y,αML) is the theoretical cumulative distribution function (cdf) for a
power law with parameters αMLand xmin= y, andˆF(x;y) is the empirical distri-
bution function (edf) over the data points with value at least y. In cases where
two values of y yield roughly equally good fits to the data, we report the one with
greater statistical significance.
Once these parameters have been estimated, we first calculate the standard error
in α via bootstrap resampling. The errors reported in Tables 1 and 2, for instance,
are derived in this manner. Finally, we calculate the statistical significance of this fit
by a Monte Carlo simulation of n data points drawn a large number of times (e.g., at
least 1000 draws) from F(x;αML,xmin), where αMLand xminhave been estimated
30 Clauset, Young and Gleditsch
as above, under the one-sided KS test. Tabulating the results of the simulation
yields an appropriate table of p-values for the fit, and by which the relative rank of
the observed KS statistic can be interpreted in the standard way.
As mentioned in the text, there are many heavy-tailed distributions, e.g., the
, the stretched exponential e−αxβ, the log-normal, and even a
different two-parameter power law (c + x)−α. For data that span only a few orders
of magnitude, the behavior of these functions can be statistically indistinguishable,
i.e., it can be hard to show that data generated from an alternative distribution
would not yield just as good a fit to the power-law model. As such, we cannot
rule out all Type II statistical errors for our power law models.On the other
hand, we note that for the distributions described in Section 4, the statistical power
test versus a log-normal model indicates that the power law better represents the
empirical data. In some sense, the particular kind of asymptotic scaling in the data
is less significant than the robustness of the heavy tail under a variety of forms of
analysis. Simply the fact that the patterns in the real-world severity data deviate so
strongly from our expectations via traditional models of terrorism illustrates that
there is much left to understand about this phenomenon, and our models need to
be extended to account for the robust empirical patterns we observe in our study.
Arce M., D. G. and T. Sandler (2005). Counterterrorism: A game-theoretic analysis. Journal
of Conflict Resolution 49, 138–200.
Bak, P. and C. Tang (1989). Earthquakes as a self-organized critical phenomena. Journal
Geophysical Research 94, 15635.
Bak, P., C. Tang, and K. Wiesenfeld (1987). Self-organized criticality: An explanation of
1/f noise. Physical Review Letters 59, 381.
Biggs, M. (2005).
American Journal of Sociology 110, 1714.
Strikes as forest fires: Chicago and Paris in the late 19th century.
Bogen, K. T. and E. D. Jones (2006). Risks of mortality and morbidity from worldwide
terrorism: 1968–2004. Risk Analysis 26, 45–59.
Cederman, L.-E. (2003). Modeling the size of wars: From billiard balls to sandpiles. Amer-
ican Political Science Review 97, 135.
Cioffi-Revilla, C. (1991). On the likely magnitude, extent, and duration of the Iraq-UN war.
Journal of Conflict Resolution 35, 387–411.
Clauset, A. and M. Young (2005).
Scale invariance in global terrorism. Preprint
On the Frequency of Severe Terrorist Events31
Congleton, R. D. (2002). Terrorism, interest-group politics, and public policy. Independent
Review 7, 47.
Crook, C. (2006). The height of inequality. The Atlantic Monthly 298, 36–37.
Enders, W. E. and T. Sandler (2006). The Political Economy of Terrorism. Cambridge:
Cambridge University Press.
Farmer, J. D. and J. Geanakoplos (2006). Power laws in economics and elsewhere. Unpub-
Bureau ofInvestigation(1999). Terrorism in theunited states.
Gartner, S. S. (2004). Making the international local: The terrorist attack on the USS Cole,
local casualties, and media coverage. Political Communication 21, 139–159.
Goldstein, M. L., S. A. Morris, and G. G. Yen (2004). Problems with fitting to the power-law
distribution. European Physical Journal B 41, 255.
Hoffman, B. (1999). Terrorism trends and prospects. In Countering the New Terrorism,
pp. 7–38. RAND Corporation.
Johnson, N. F., M. Spagat, J. Restrepo, J. Bohorquez, N. Suarez, E. Restrepo, and
R. Zarama (2005).From old wars to new wars and global terrorism.
Johnson, N. F., M. Spagat, J. A. Restrepo, O. Becerra, J. C. Bohorquez, N. Suarez, E. M.
Restrepo, and R. Zarama (2006). Universal patterns underlying ongoing wars and ter-
rorism. Preprint arxiv.org/abs/physics/0605035.
Kleiber, C. and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial
Sciences. New Jersey: John Wiley & Sons, Inc.
Lacina, B. A. (2006). Explaining the severity of civil wars. Journal of Conflict Resolution 50,
Li, L., D. Alderson, R. Tanaka, J. C. Doyle, and W. Willinger (2006). Towards a theory
of scale-free graphs: Definition, properties, and implications. Internet Mathematics 2,
Li, Q. (2005). Does democracy promote or reduce transnational terrorist incidents? Journal
of Conflict Resolution 49, 278–297.
Malamud, B. D., G. Morein, and D. L. Turcotte (1998). Forest fires: An example of self-
organized critical behavior. Science 281, 1840.
Mickolus, E., T. Sandler, J. Murdock, and P. Fleming (2004). International terrorism:
Attributes of terrorist events 1968-2003(ITERATE). Dunn Loring, VA: Vinyard Software.
Mitzenmacher, M. (2004). A brief history of generative models for power law and lognormal
distributions. Internet Mathematics 1, 226.
National Memorial Institute for the Prevention of Terrorism (2006). Terrorism knowledge
Navarro, P. and A. Spencer (2001). September 11, 2001: Assesing the costs of terrorism.
Milken Institute Review: Fourth Quarter 2001, 16–31.
Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf’s law. Contemporary
Physics 46, 323.
32Clauset, Young and Gleditsch
Overgaard, P. B. (1994). The scale of terrorist attacks as a signal of resources. Journal of
Conflict Resolution 38, 452–478.
Pape, R. A. (2003). The strategic logic of suicide terrorism. American Political Science
Review 97, 3.
Press, W. H., S. A. Teukolsy, W. T. Vetterling, and B. P. Flannery (1992). Numerical
Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press.
Reed, W. J. and B. D. Hughes (2002). From gene families and genera to income and internet
file sizes: Why power laws are so common in nature. Physical Review E 66, 067103.
Reich, W. (1990). Origins of Terrorism. Cambridge: Cambridge University Press.
Richardson, L. F. (1948). Variation of the frequency of fatal quarrels with magnitude.
Journal of the American Statistical Association 43, 523.
Rosendorff, B. P. and T. Sandler (2005). The political economy of transnational terrorism.
Journal of Conflict Resolution 49, 171.
Sandler, T. and D. G. Arce M. (2003). Terrorism and game theory. Simulation & Gaming 34,
Sandler, T. and H. Lapan (2004). The calculus of dissent: An analysis of terrorists’ choice
of targets. Synthese 76, 245–261.
Serfling, R. (2002). Efficient and robust fitting of lognormal distributions. North American
Actuarial Journal 6, 95.
Shubik, M. (1997). Terrorism, technology, and the socioeconomics of death. Comparative
Strategy 16, 399.
Simon, H. A. (1955). On a class of skew distribution functions. Biometrika 42, 425.
United States Department of State (2003). Patterns of global terrorism.
Wilkinson, P. (1997). The media and terrorism: A reassessment. Terrorism and Political
Violence 9, 51–64.
Zipf, G. (1949). Human Behavior and the principle of least effort. Cambridge: Addison-