# Why do Terrorist Attacks Satisfy a Scaling Law?

Alvin M. Saperstein

Abstract: It has been empirically well verified that the relation between the severity of a terrorist act, as measured by fatalities or casualties produced, and the frequency of these acts, satisfies a decreasing power law – a scaling law. Attempts to explain this law have focused on the behavior and structure of the terrorist cells. This paper points out that the postulates upon which these explanations are based have little empirical validity. However, applying the same derivation to the dynamic organization of both terrorist and the victim populations leads to the same power law without the previous discrepancy between postulate and reality.

Situations in which the probability, or frequency, of the occurrence of an event is proportional to a power of that event are said to satisfy a "scaling law". Concisely, if f is the frequency and x the magnitude of the event as measured by the numbers of casualties produced by the event, f~x-a, where the exponent "a" may be either a positive or negative number. It is called a scaling law because if the magnitude of each event is multiplied by a constant number k, the frequency is just multiplied by the constant factor k-a, independent of the value of x. Situations in which the observed frequencies of possible events satisfy such a scaling law over large ranges of x are very common in the physical and biological sciences [1]. For example, if x represents the population of American cities and f the number of cities having such a population, f satisfies a power law with exponent ~2; similarly, the frequency with which earthquakes of magnitude x occur roughly satisfies a power law with exponent ~2/3. Such laws have also been empirically found to accurately describe many situations of interest to the social sciences [2]. Richardson, analyzing many wars, was able to show that the frequency of wars resulting in x fatalities could be accurately expressed as f~x-a, where a is a positive number [3]. More recently, a number of studies have been made of the frequency of terrorist attacks resulting in x casualties in civil conflicts throughout the world, and they have been found to very accurately satisfy such a scaling law with a value for a in the vicinity of 2.5[4].

When an event is the result of many small independent causes, it is usually expected to satisfy a normal distribution (also referred to as a "Gaussian" or a "bell curve"). In such a distribution, outcomes are closely bunched about the mean value. For example, if one-half of all possible events are expected to occur in an interval of width 2σ centered upon the mean value <x> of the normal distribution, then 0.82 of the events are expected to occur in the interval when its width is doubled, i.e., only 30% more. On the other hand, if the distribution follows a scaling law with positive exponent, and if σ << <x> (not a necessary requirement for either Gaussian or power law distributions), doubling the width of the interval about the mean leads to a doubling of the number of events included (100% more). Thus, scaling laws with positive exponents are said to have "a very heavy tail". It is expected that if such distributions govern the process, many events will occur very far from the mean.

To illustrate the heavy tail, the distribution of worldwide deaths per terrorist attack since 1968 has been shown to have a mean value of ~4 and a standard deviation of ~32 [5]. If these deaths were distributed normally, the probability of a terrorist incident resulting in ~ 100 deaths (3 standard deviations) would be 7 x 10-5 – a very unlikely event. Empirically, the heavy tail leads to a thousandfold greater probability that there would be 100 or more deaths, a fact very evident to readers of current newspapers.

Given the apparent universal validity of the power law as an empirical representation of the effects of terrorist acts, it is natural to seek an explanation of the law. Such an explanation would go a long way to satisfy inherent human intellectual curiosity. It might also furnish tools for action – either by the forces of "law and order" to diminish the effects of terrorist acts, or by the terrorist groups seeking to enhance their effectiveness. Several quite different theoretical models leading to scaling laws have been suggested in the references already given, but it is hard to believe that any one model will adequately explain the varied applications of scaling laws. For example, some attribute innovation in cities to the network connections between the various social factions in a city that maintain the cohesiveness of that city [2]. Such a model, however, is very unlikely to be valid for terrorist groups trying to destroy social cohesiveness. Thus, it is not expected that a single type of explanation would serve for all examples of power law distribution.

Papers that try to explain the terrorism scaling law focus on the behavior of the organizations of the terrorists [4, 5, 6]. In their "toy model", Clauset, Young and Gladitsch focus on the behavior of the terrorist movement, as a single entity, planning and carrying out a single attack [5]. They postulate that the probable severity of such an attack increases with the time spent planning the attack so that the rate of increase of this probability is itself proportional to the probability. From this postulate, it follows that the probable severity increases exponentially with time: p(t) ~ekt, k>0. They also postulate that the protective organs of the society are also planning to minimize the effects of terrorist actions so that the rate of decrease of probable severity is proportional to the magnitude, x, of the expected severity. Hence, the probable severity is a decreasing exponential in time: x ~ eλt,λ < 0. Combining these two assumptions, we obtain the desired power law: P (x) ~ xa, where a = 1 - k/λ [1].

However, it is difficult to reconcile what we know of reality with the general validity of the two postulates. It may very well be that the success of an outlier event, producing casualties in the thousands, like the September 2001 attack on the World Trade Center, required a long interval of planning. But if one looks at events that caused fatalities ranging from one to one hundred, there is scant evidence that the suicide bombing that kills fifty people in a town market place required a much greater planning time than the attack on a military patrol that kills a few non-combatant bystanders. And there is certainly little support for the second postulate. The "authorities" certainly spend more time trying to guard against the multiple few-fatality events expected in a "combat zone" than they do against unexpected outlier events in non-combat zone such as New York City. Given the weakness of its two postulates, the apparent success of the double exponential mode, in deriving a scaling law, must be just fortuitous.

A different organizational model of the severity of terrorist attacks, not depending upon planning time, is based upon the postulate that the severity of damage inflicted in such an attack is proportional to the size of the terrorist "cell" carrying out the attack [4, 6]. This model assumes that the terrorist movement is made up of cells whose size is constantly changing. Small cells come together in random encounters, aggregating into larger cells. Larger cells randomly disintegrate into smaller cells, either because of internal conflicts, satisfaction with a completed "job", or pressure from police and military forces. The mathematical model is that of a "master equation" in which the rate of change of the number of groups of any given size (the rate of change of the "frequency" of that size) is proportional to a sum of the products, two by two, of the frequencies of all other sized groups (determining the probability of aggregation) and a linear sum over all of the frequencies (giving the probability of disintegration of the cells) [1]. The resultant non-linear, time-dependent, set of differential equations cannot be solved in general. And so another postulate is introduced – that the time variation can be ignored. It is assumed that after some unspecified time, the world of the terrorists, the society upon which they prey, and the military and police forces trying to protect that society, has reached a steady state so that on average, none of the cell-size frequencies are time-dependent. The resultant time-independent master equation can be solved, and, after some reasonable approximations, gives the result that the frequencies of cell size (the number of terrorist cells having a specified number of members) satisfies the power law with the desired exponent equal to 5/2. Using the postulated proportionality between cell size and severity of damage inflicted by a given cell in any given attack, one obtains the desired empirically observed scaling law, in which the frequency f of an attack producing x casualties is proportional to a negative power of x.

Again, it is difficult to accept the empirical validity of the two postulates. Terrorism is a time-dependent process, starting with the present state of a society and ending, sooner or later, with either the triumph of the terrorists and a major change in the society, or with the defeat of the terrorists and the preservation (in some form) of the status quo [7]. In either final steady state there will be no frequency of terrorist-cell size to satisfy a power law. Either the terrorist cells will all voluntarily disband with the triumph of the terrorists, or they will be quashed by the victory of the status quo forces. Furthermore, there is little to support proportionality between casualty rate in a terrorist attack and the number of terrorists in the cell carrying out the attack. Massive terrorist attacks often result in few casualties among the target population, whereas single "lone wolf" suicide attacks (cell size of unity) have often resulted in many fatalities and injuries. So a model just built upon the cell sizes of the terrorists cannot be very satisfactory.

The difficulty created by the second postulate – that the damage inflicted by a terrorist cell is proportional to the size of that cell - can be removed by considering the role of the victims in determining the severity of a terrorist attack. The members of a civil society, the non-combatants, who are the presumed targets of the terrorist attacks, also are members of groups of varying and fluctuating sizes. There are familiar, friendship "cells", groups who go to market together, who pray together, work together, play sports together, engage in political or cultural events together. It is easy to presume that the frequency of cell size, for these "civilian" groups, also satisfies a similar master equation having linear and bilinear terms. Again presuming time-independent steady state solutions, the size frequency of these civilian cells should also satisfy a similar power law with the same exponent. One could then postulate a proportionality between size of victim group and the number of casualties inflicted upon that group by a terrorist attack to again derive the observed desired casualty-frequency scaling law.

But there is no more reason to believe in a proportionality relation between victim-cell size and severity of attack than to believe in a proportionality between terrorist-cell size and attack severity. A victim-cell proportionality would imply a tight correlation between the number of casualties inflicted in an attack upon a city and the population of that city. But footnote 9 of the Clauset, Young and Gleditsch paper, shows that there is a very weak association between these two variables [5].

However, the damage inflicted in a terrorist attack is not likely to depend only upon the number of attackers or only upon the number of targets. The damage is more likely proportional to the product of the two numbers [8]. Certainly, the losses inflicted will depend upon the number of those firing (the terrorist cell size). But the terrorists are unlikely to be aiming at specific individual targets; more likely, they are aiming at a general area and the probability of a hit depends upon the number of targets in the area (the number of potential victims in the target group, a group whose size distribution is determined by the same form of master equation governing the terrorist cell size distribution). Thus the number of casualties inflicted is likely to be proportional to the product of the two cell sizes [9, 10]. Since the two cell sizes, terrorist and victim, satisfy the same distribution law, with the same exponent, it follows that their product also satisfies that same frequency distribution with the same exponent.

Thus, bypassing the conceptual difficulties of the steady-state (time independent) hypothesis, we have an "explanation" of the empirical terrorist attack severity- frequency law. One policy implication for minimizing casualties inflicted by terrorists seems immediate: make sure that neither the terrorist cells, nor the potential victim cells, grow to large sizes. (Controlling "victim cell size" in most societies may be very unrealistic.) Have your defense forces strive to their utmost to fragment the terrorist cells and discourage the massing of potential victims in vulnerable areas – such as market places. This suggestion certainly comports with "common sense". Another suggestion is directed at the steady state hypothesis. Since a time independent situation leads to the observed scaling law, perhaps a policy directed at forcing time dependence will change the severity-frequency relation, hopefully for the better a decrease in casualties. Such a policy implies fast, continuous reactions by the defending forces, never allowing the terrorist cell structure to relax to a steady state. This suggested policy again seems to agree with common sense, whether or not it is commonly put into practice.

It thus appears, so far, that a mathematical derivation of the scaling law, a theoretical understanding of the immediate effects of terrorist victims, does not lead to novel policy suggestions for action. However, a policy for action is more likely to receive public support if it is based upon multiple, different, theoretical and empirical supports – in this case mathematical as well as the usual verbal conceptualizations. (After all, any political candidate usually tries to say as many good things about himself, and as many bad things about his opponent as possible, to convince the public to vote for him – a policy decision.) Science is like a woven cloth; its strengths increase with the number of differently oriented threads passing through, and supporting, each point [10]. There is much that the public policy-maker can learn from the normative processes of science.

References
1. Newman, M.E.J. (2005) Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46, 323.
2. See, for example, Arbesman, Samuel, Jon M. Kleinberg, and Steven H. Strogatz (2009) Superlinear scaling for innovation in cities. Physical review E, 79, 016115.
3. Richardson, L.F. (1960) Statistics of Deadly Quarrels, eds., Q. Wright and C.C. Lienau, Boxwood Pres, Pittsburgh, PA.
4. Johnson, N.F., M. Spagat, J.A. Restrepo, O. Becerra, J.C. Bohorquez, N. Suarez, E.M. Restrepo, and R. Zarona (2006) Universal patterns underlying ongoing wars and terrorism. Preprint: arXiv.org/abs/physics/060535.
5. Clauset, A., M. Young, and K.S. Cleditsch, (2007) On the frequency of severe terrorist events. Journal of Conflict Resolution, 51 (1) 58-87.
6. Clauset, Aron and Frederik W. Wiegel (2010) A generalized aggregation-disintegration model for the frequency of severe terrorist attacks. Journal of Conflict Resolution, 54 (1) 179-197.
7. Saperstein, A. M. (2010) A mathematical model of suicide terrorism, in Adrienne M. Gallore, ed. Terrorism: Motivation, Threats and Prevention. Hauppauge, N.Y.: Nova Science Publishers.
8. Lanchester, F.W. (1914) Aircraft in Warfare: The dawn of the Fourth Arm – No. 5, The Principles of Concentration. Engineering 98, reprinted in World of Mathematics, James R. Newman, ed., Vol. IV, 2138-2159, New York, N.Y.: Simon and Schuster, 1956.
9. Taylor, James G. (1983) Lanchester Models of Warfare. Arlington, VA, Operations Research Society of American, Military Applications Section, 2 volumes.
10. Saperstein, A. M. (2008) Mathematical Modeling of the Interaction Between Terrorism and Counter-terrorism and Its Policy Implications. Complexity, 14 (1) 45-49.
Alvin M. Saperstein
Professor of Physics
Department of Physics and Center for Peace and Conflict Studies
Wayne State University
Detroit, MI 48202

These contributions have not been peer-refereed. They represent solely the view(s) of the author(s) and not necessarily the view of APS.