The Power Law of Terrorism

Research result #1: “A Generalized Fission-Fusion Model for the Frequency of Severe Terrorist Attacks,” by Aaron Clauset and Frederik W. Wiegel.

Plot the number of people killed in terrorists attacks around the world since 1968 against the frequency with which such attacks occur and you’ll get a power law distribution, that’s a fancy way of saying a straight line when both axis have logarithmic scales.

The question, of course, is why? Why not a normal distribution, in which there would be many orders of magnitude fewer extreme events?

Aaron Clauset and Frederik Wiegel have built a model that might explain why. The model makes five simple assumptions about the way terrorist groups grow and fall apart and how often they carry out major attacks. And here’s the strange thing: this model almost exactly reproduces the distribution of terrorists attacks we see in the real world.

These assumptions are things like: terrorist groups grow by accretion (absorbing other groups) and fall apart by disintegrating into individuals. They must also be able to recruit from a more or less unlimited supply of willing terrorists within the population.

Research Result #2: “Universal Patterns Underlying Ongoing Wars and Terrorism,” by Neil F. Johnson, Mike Spagat, Jorge A. Restrepo, Oscar Becerra, Juan Camilo Bohorquez, Nicolas Suarez, Elvira Maria Restrepo, and Roberto Zarama.

In the case of the Iraq war, we might ask how many conflicts causing ten casualties are expected to occur over a one-year period. According to the data, the answer is the average number of events per year times 10^-2.3, or 0.005. If we instead ask how many events will cause twenty casualties, the answer is proportional to 20^-2.3. Taking into account the entire history of any given war, one finds that the frequency of events on all scales can be predicted by exactly the same exponent.

Professor Neil Johnson of Oxford University has come up with a remarkable result regarding these power laws: for several different wars, the exponent has about the same value. Johnson studied the long-standing conflict in Colombia, the war in Iraq, the global rate of terrorist attacks in non-G7 countries, and the war in Afghanistan. In each case, the power law exponent that predicted the distribution of conflicts was close to the value 2.5.

This doesn’t surprise me; power laws are common in naturally random phenomena.

Tags: academic papers, Iraq, risk assessment, terrorism

Posted on January 12, 2010 at 1:46 PM • 25 Comments

Comments

Murk • January 12, 2010 1:59 PM

10^2.3 is closer to 200 than 0.005. I suspect you (or the original) meant 10^-2.3

It doesn’t make much sense to say ‘proportional to a constant’ anyway. It looks like what’s implied is that

number of events is proportional to number of casualties in event to the power of -2.3

Nevertheless, an interesting pattern

Dave Aronson • January 12, 2010 2:14 PM

Murk, your analysis is correct, the minus sign on the exponent got dropped. That’s an artifact of extracting text from one place to another. Good thing Bruce provided links….

Anurag • January 12, 2010 2:39 PM

result #1 would seem to point to the higher effectiveness of disrupting cells (through infiltration, bombings, etc) than in safeguarding targets (airport screening, etc).

JP • January 12, 2010 2:39 PM

Yes, he’s saying that the probability P that an event kills N people is P(N) = N^-k, where k = 2.3 (for Iraq).

kangaroo • January 12, 2010 2:41 PM

Power laws are common for non-independent random systems, while normal distributions are common for independent random systems.

Almost everything is non-independent — but we usually use independent statistics because they’re simpler. Very bad assumption that a system that is simpler for us, is in general accurate.

It’s the same crap that messed up the economy — thinking that an assortment of independent events is “close enough”.

Brandioch Conner • January 12, 2010 2:59 PM

@JP
Nice math. Do you mind if I put some numbers in it?

In Iraq, a successful attack will kill (injure?) …
1 person 100% of the time.
2 people 20% of the time.
3 people 8% of the time.
4 people 4% of the time.
5 people 2% of the time.
6 people 1% of the time.
7 people 1% of the time.
8 people .8% of the time.
9 people .6% of the time.
10 people .5% of the time.

Any strangeness is probably due to rounding.

Greg • January 12, 2010 3:13 PM

On the second research result, there is an interesting Ted talk by a different researcher, Sean Gourley, on “The Mathematics of War” that tries to argue that moving away from that 2.5 constant is key to ending the conflict.

http://www.ted.com/talks/lang/eng/sean_gourley_on_the_mathematics_of_war.html

Tom • January 12, 2010 3:50 PM

“And here’s the strange thing: this model almost exactly reproduces the distribution of terrorists attacks we see in the real world.”

It’s “strange” that a model hypothesized from the data matches the data? I would only consider it “strange” if the model predicts observations other than the ones to which the model’s “simple assumptions” were fit. (And maybe it does; I didn’t dig into the paper yet.) It always bugs me when science writing misses that distinction, though.

John K • January 12, 2010 4:07 PM

I am loathe to attribute a power law distribution from the empirical data of anything, without, oh I don’t know, making sure the data was analyzed properly? Yeah, that’s the ticket.

http://cscs.umich.edu/~crshalizi/weblog/491.html

David • January 12, 2010 4:39 PM

How does the fatality law work on the tails? We’ve had a good many actions that caused over a million deaths last century, including the Nazi genocides and two world wars just for starters. A million to the -2.3 power is roughly 10^-14. If you think there were maybe 5 megakills last century (and I can count more), that would indicate about five hundred trillion individual murders, which seems excessive.

kangaroo • January 12, 2010 5:50 PM

JohnK:

That’s at least better than the common assumption of normal distribution for systems without distributions. Pick up almost any molecular biology paper — odds are that there’ll be five graphs that are treated as normal that you can tell by the naked eye aren’t normal, and two graphs that aren’t normal by their definition.

Yup, it’s always fun when two supposedly normally distributed variables are divided, and the ratio is treated as normal, or when the curve butts into a wall — and it’s still treated as normal.

That’s how you get your medicines, as a matter of fact!

Peter E Retep • January 12, 2010 7:14 PM

There are old stastics about homicide loss rates that are reported to be roughly constant in war or not – – -for a given society.

Is it possible that the power function is exactly that,(?) a consequence of the amplification of individual’s power to access information and connect the dots due to our internet and media infrastructure, and their ability to find like-thinking persons to associate with?

Clive Robinson • January 12, 2010 7:30 PM

@ kangaroo,

” …or when the curve butts into a wall — and it’s still treated as normal.

That’s how you get your medicines, as a matter of fact!”

Yup when I was a semi active sports person I used to get my medicines after curving and butting into a wall 8)

On a more serious note.

Is this due to resourcing issues?

If you take a group of people and ask them to do a specific task independently of each other then it takes them an average period of time.

If however you put them into groups of two and ask them to the same task it takes a little bit longer than half the average period for a single person.

The more people involved the worse the problem, often this is put down to “communications” or mental capacity (5 + or – 2) issues.

However it follows a power law.

It is one reason that large projects work better the more independent the teams are and the smaller the teams are. However the tasks each team does needs to be likewise smaller or less complex.

Also the amount of “in fighting” and meetings in projects can be kept considerably lower in small team sizes (and the 5 + or – 2 rule for teams applies).

On another note as Bruce probably knows “cooking times” for meat tend to follow a power law although it is such it apears (aproximatly) linear to most people ie 25min/pound + 25min for the oven.

Clive Robinson • January 12, 2010 7:48 PM

@ Peter E Retep,

“Is it possible that the power function is exactly that,(?) a consequence of the amplification of individual’s power to access information”

The old “force multiplier” effect.

The question is why for war does it pan out to N^-2.5 (or there abouts).

You would expect a physical force limits to be effectivly N^-2 that is effort/area at a distance.

David Thomas • January 12, 2010 8:27 PM

@John K

According to my (admittedly limited) understanding, the analysis described in the paper Bruce linked to more or less correspond in content, if not form, with those proposed by the paper you linked to (well, the paper the rant you linked to was based on). Did I misread?

In particular, which of these are you claiming?

1) You subjected the data from the study to the analysis you are advocating, and it does not seem to fit a power-law distribution.

2) The paper, for lack of subjecting the data to appropriate analysis, does not show that the data fits a power law distribution.

3) The blurb which Bruce posted does not contain sufficient information to determine whether the statistical analysis presented in the paper was done well.

Home Security Cameras • January 12, 2010 10:48 PM

The findings by Professor Neil Johnson are very interesting. Remarkable in fact. An eye-opener for me; I didn’t know power laws would apply this way in these situations.

Aaron • January 12, 2010 11:58 PM

You might also check out

On the Frequency of Severe Terrorist Attacks, by A. Clauset, M. Young and K. S. Gledistch. Journal of Conflict Resolution 51(1), 58 – 88 (2007).

More generally, all of this stuff, on wars, insurgencies and terrorism, does connect with L.F. Richardson’s original work on wars and homicides. In a sense, it’s the extension of that work, but using better data and, in some cases, better analytic methods. I don’t think the modeling side of things has been answered yet though. There are some interesting ideas, but there’s a lot of work left to validate and test them.

Sean • January 13, 2010 3:19 AM

Interesting post and thanks everyone for the comments here. As it happens my research group released a new paper on this topic in last month’s issue of the science journal Nature.

‘Common ecology quantifies human insurgency’ Bohorquez, Gourley, Dixon, Spagat & Johnson.

http://www.nature.com/nature/journal/v462/n7275/full/nature08631.html

This research paper moves on from the earlier work done by Gourley and Johnson. In the paper we uncover a new set of results not only in the size distribution of conflict events but also in the temporal distribution of attacks. The paper presents a unified model of human insurgency that agrees with the empirical evidence collected from 9 different wars and over 54,000 unique events.

you can read more about this work over at the website

http://www.mathematicsofwar.com

on this site you can download copies of the Nature paper as well as the more detailed supplementary info to accompany it. In the supplementary information (see here http://bit.ly/7twWaU) you can read about where the data comes from and what different types of information sources are used for the analysis.

If mathematics is not your preferred language, then you can read a more descriptive account of the research over at the TED blog where they have an in depth interview with Gourley http://bit.ly/87DoWK

Clive Robinson • January 13, 2010 6:55 AM

@ Sean,

I had a quick look at the TED talk, and I was struck by a couple of things,

First most notably, that the alpha changes with time.

Secondly,

“When a small/weak group of people take on a much stronger opposition, they have to find an organizational structure and strategy that will allow them to compete. It turns out that there are only a small number of possible solutions to this problem, and if an insurgent group does not adopt one of these solutions, they generally do not survive.”

The first begs the question of “human” -v- “technology” that is are the human aspects causing the change in alpha or is it technology and it’s “force multiplier” effect and “action at a distance” issues.

That is, is the change in alpha tied to say communications or transport or other tangable force multiplier alowing a greater physical coverage area for each individual element in a conflict. Or is it due to an intangable element such as better Intel gathering / processing.

If it is tied to tangable resources then this gives a reliable method of reducing the effectivness of insergents et al. If it is due to intangable resources then like an air bubble under freshly put up wallpaper where ever you press you move the problem not realy remove it (yup weak analagy alert 😉

Which brings me onto the second thing “organisational structure” you note that there are only a few structures that (currently) work successfully.

Thus a couple of questions arise,

Firstly can the insurgents use this information to their advantage (a case of the experiment being changed by the observation process). Such a thing has been seen in share trading with prediction models to place trades as oposed to assess a trades risk.

The second is does it alow the majority forces to make stratigic efforts to disrupt such organisational structure and thus increase the fragmentation rate at the bottom.

That is instead of just attacking the “tallest Poppy” throw a spanner in the works of stronger groups by fragmenting small groups to overwhelm the larger groups with internal organisational issues.

Thus a group that is successfull could easily be bogged down by it’s own success. As the fragments seek to enlist they need to be checked out by the group they wish to join, this can take considerable resources that would otherwise be devoted to other asspects. Further if overloaded in the right way it will alow for assets to be placed or recruited in supply chains etc.

dragonfrog • January 13, 2010 10:38 AM

@Anurag “result #1 would seem to point to the higher effectiveness of disrupting cells (through infiltration, bombings, etc) than in safeguarding targets (airport screening, etc).”

The bombing part is true if and only if:

1) you are able to control the definition of terrorism, such that bombings of civilian targets committed by your side are “targeted strikes” or the like, but only the bombings of civilian targets committed by the other side are “terrorism”.

2) your campaign of bombing civilian targets (let’s leave the T word out of this one) is more effective at killing groups of your enemies, than it is at bolstering your enemy’s recruitment efforts.

HJohn • January 13, 2010 10:57 AM

@dragonfrog: “your campaign of bombing civilian targets (let’s leave the T word out of this one) is more effective at killing groups of your enemies, than it is at bolstering your enemy’s recruitment efforts.”

I believe this is a deliberate strategy by the T’s. They know that if they put innocents in harms way, that any response to their aggression will result in innocents getting hurt, which makes for pretty good propaganda for them.

It would be naive not to think they do not know this and leverage it to their advantage.

There are many good people who disagree about strategies, but I for one am thankful that I’m not responsible for dealing with it. Seems doing nothing makes it worse, and responding makes it worse. There isn’t a solution that I can think of that isn’t hell.

paul • January 13, 2010 11:52 AM

I’m a little taken by the notion that terrorist groups grow by accretion and shrink by splitting into individuals. This reverse what one might think of as the common wisdom (perhaps mostly acquied through fiction) that larger groups recruit individuals and shrink by splintering into factions.

Or perhaps the behavior works that way even if the actual “membership” stats don’t.

John • January 13, 2010 12:00 PM

@HJohn:

My solution would be to debase their moral/ethics structure to preoccupy their society with sex. These areas are, after all, typically under secular control (fancy way to say “religious” but…), with porn and prostitution and the like banned.

When you make love you’re using up energy; and afterwards you feel happy and don’t give a damn for anything. They can’t bear you to feel like that. They want you to be bursting with energy all the time. All this marching up and down and cheering and waving flags is simply sex gone sour.

Remove the legal punishments for misbehaved women in tight tube tops and short skirts, and the lecherous desires of men will take over. And then who would the terrorists recruit? Why should I blow myself up when I’m constantly getting laid by college cheerleaders in micro-miniskirts and knee-high-boots?

That’s what I think. There’d be no more marching and cheering and rallying to blow up America if everyone was too busy figuring out how they’re going to get away with screwing their neighbor’s high school daughter. That’s how Americans get by day to day without blowing themselves up.

Clive Robinson • January 13, 2010 2:43 PM

@ HJohn,

“I believe this is a deliberate strategy by the T’s. They know that if they put innocents in harms way, that any response to their aggression will result in innocents getting hurt, which makes for pretty good propaganda for them.”

Civilian shields is a two way street.

For instance a certain nation just over a year ago invaded the land of another nation that they had repeatedly terrorised in one way or another.

The invading nation where caught out firing white phosphorus shells into an area the held a UN compound. The premier of the invading country told the UN in a face to face meeting that T’s where firing at his invading troops from within the UN compound.

This was shown to be absolutly untrue. The military of the invading nation then claimed that shots had been fired from behind the UN compound.

Film actually being recorded at the time the invading nation started shelling the UN compound humanitarian aid showed that this also was a compleate load.

Therefore I think it fairly safe to claim that atleast one nation makes deliberte attacks on what is clearly humanitarian aid.

Or more emotivly uses tactics virtualy straight out of the “Nazi Propagander Playbook” to hide their deliberate “ethnic cleansing” policy in an adjacent nation.

By the way emotive words or not the invading nation concerned has a long history of trying to hide it’s “land grab” policies behind decite, it even very deliberatly attacked a US (NSA) vessel with MTBsto try and prevent the deciet coming out untill their land grab objectives had been forfilled.

The then POTUS decided not to publicaly award galantry and other valour meddles to the crew of vessel incase it lost him votes…

Such is the twisted little nature of politics and claims about those against whom illegal wars are being prosecuted.

spy store • December 21, 2010 12:49 AM

ok, after reading greg’s comment and resource http://www.ted.com/talks/lang/eng/…, where in all the mathematics is the variable of the type of enemy we are fighting and considering into the calculation. Meaning – recent wars of late are dealing with terrorist groups (cells) as opposed to an organized army, with which we could annihilate much easier today if our enemy was an organized army! Those are numbers for thought!

The Power Law of Terrorism

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