The Mathematical Structure of Terrorism

[toughenough]

Seems a little out there to me, but interesting nonetheless...curious to see the discussion on this one.

http://www.physorg.com/news67524254.html

The Mathematical Structure of Terrorism
By Ben Mathiesen
May 22, 2006

The complex patterns of the natural world often turn out to be governed by relatively simple mathematical relationships. A seashell grows at a rate proportional to its size, resulting in a delicate spiral. The gossamer network of galaxies results from the simple interplay between cosmic expansion and the force of gravity over a wide range of scales. As our catalogue of natural phenomena has grown more complete, more and more scientists have begun to look for interesting patterns in human society.

The nature of war is a question of great interest to everyone, especially as the era of large-scale conflicts recedes into the past. The wars of today tend to be lopsided affairs, where guerilla forces, insurgent groups, and terrorists oppose incumbent governments. Instead of a few large-scale battles, this situation leads to an apparently random series of small-scale attacks against vulnerable targets of opportunity.

While affected governments collect records of past attacks, the random nature of such wars means that these data are of limited use in predicting future attacks. When classified according to their frequency and intensity, however, the events of any insurgent war appear to follow a power law. It should come as no surprise that weaker attacks are more common than stronger attacks, but a power law distribution makes a much more specific prediction. It turns out that if individual conflicts (for example, a terrorist attack or a guerilla raid) are classified according to the resulting number of fatalities n, then the number of such conflicts occurring in any given year is proportional to n raised to a constant power.

Let’s look at a specific example. 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.

What’s more, in the case of Colombia and Iraq he was able to show that the exponent seemed to be evolving towards that value; Colombia from above, and Iraq from below. Does this hint at a simple underlying pattern driving the behavior of modern wars?

Johnson thinks so, and has even developed a model that predicts a power law distribution of casualties with the correct exponent. In his model, the insurgent force consists of a fixed number of attack units (a general term which may include equipment or even information, as well as people) which may group together to form larger units. Each unit on its own is assigned a ‘strength’ of one, meaning that a conflict involving that unit will result in one death. Coalitions of units pool their strength, and cause proportionally more deaths.

The key ingredient in this model is the evolution of groups over time. Terrorist organizations, for example, typically function in relatively small units. When an opportunity comes up that demands more resources, they may band together. When the authorities grow too close for comfort, on the other hand, they may split up. In time these competing pressures can create a stable arrangement of groups, with a fixed distribution of different sizes.

Johnson’s model adopts a very simple dynamic to model this evolution. In any given time step, one group of attack units is randomly chosen. Each group's chance to be chosen is proportional to its size, but the many small groups still see much more activity than the few large groups. The group selected is given a small probability (1%) of disbanding into individual units; if it doesn’t disband, then it joins up with another randomly chosen group.

These are the only rules of the model, and they turn out to work just fine. After the population is allowed to evolve for a long time, the result is a power law distribution of group sizes with an exponent of exactly –5/2. Since group size is proportional to attack strength, this distribution also predicts the frequency of attacks causing a given number of fatalities. It is also interesting that the result of this model depends only on the probability of fragmentation. As long as this probability is reasonably small, the distribution of attacking groups will settle into a steady state with a power law distribution.

Is this new ‘Law of Terrorism’ really universal? “Power law patterns will emerge within any modern asymmetric war being fought by loosely organized insurgent groups.” Johnson speculates, “Although future wars will provide the ultimate test.” Johnson’s research continues with the analysis of data from other conflicts, such as Senegal, Indonesia, Israel, and Northern Ireland.

Citation: Neil Johnson et al. 2006, “Universal Patterns Underlying Ongoing Wars and Terrorism”, http://xxx.lanl.gov/abs/physics/0605035

By Ben Mathiesen, Copyright 2006 PhysOrg.com

 

[Centurian1985]

Wow - I went to the source document and most of this stuff is high level mathematics that I have no idea what they're talking about!

http://xxx.lanl.gov/ftp/physics/papers/0605/0605035.pdf

Found this to be the only summary line of interest:

"War in Iraq. At the beginning of the war in 2003, the power-law exponent was quite low and was essentially the same value as conventional wars. This is consistent with the war being fought by a conventional Iraqi army against the Coalition forces. There is then as break in this value after a few months (i.e. the conventional war ended) and following this the power-law exponent gradually rose towards 2.5. This suggests that the insurgents have been increasingly favoring more temporary attack units, with an increasingly rapid fragmentation-coalescence process. This finding could be interpreted as being a result of increased success by the Coalition Forces in terms of forcing the insurgents to fragment.  On the other hand, it also means that the Iraq War has now moved to a value of α, and hence character, which is consistent with generic non-G7 terrorism and it may therefore be hard to make further progress against them."

Seems to imply that based on their calculations, the military forces in country are now at a standstill with the insurgent forces.

Disagree with the title - they appear to be seeking a method of attempting to predict the severity and number of insurgent acts rather than defining the structure of terroism.

 

[Thirstyson]

I think the title is fine, they only define the Mathematical structure. They are simply doing statistical analysis on terrorism data.

Still, the conclusions are interesting.

 

[Pieman]

All you need to understand this paper is this:

http://mathworld.wolfram.com/LeastSquaresFittingPowerLaw.html

HA! Just kidding. There is a little more going on here.

I think the title is fine, they only define the Mathematical structure. They are simply doing statistical analysis on terrorism data.

Not quite, they have constructed a rather simple stochastic model outlined in detail on page 22. What is interesting is they claim their model outputs similar patterns of behaviour to real world situations. I think it could be an interesting step towards understanding asymmetric warfare. They are most certainly relating their model to real world situations way too much though. It is not something you would bet money on as their model is way too simplistic.

I will have to read through this paper on more detail later, kinda interesting.

 

[Centurian1985]

I've heard of some departments trying to create predictive models for some time.  None were considered successfull. Does this look like a 'final answer' to either of you or only another step towards a final answer?

 

[DBA]

Modeling a few variables that underlie a complex system can give some insight but aren't really predictive. That's because complex systems tend to have other variables that can dominate at any time.

 

[Long in the tooth]

The underlying argument here is fine as it relates to frequency versus size of attacks.  The very nature of terrorism is based on the comparitive advantage of secrecy and surprise.  These two factors are of course countered by vigilence by the armed forces and individuals.  Terrorists banding together in groups to create resources capable of defeating the enemy are also easier to detect and smash as they increasingly resemble conventional forces with the C3I and logistics required.  Deception plays a vital role but is hampered as it is a force multiplier and higher function not easily acquired, and vulnerable to both conventional attacks and intelligence infiltration.

An equilibrium exists.  I would suggest that the constant is based on the amount of external support available in order to purchase weapons, pay members, give gratuities to families of casualties and of course bribe local officials.  We should see differences when we look at Viet Nam (Genl Giapp sp by Russia, North Viet Nam and China), the IRA (sp by US individuals), and Afghanistani terrorists.  As the amount of external sp decreases the probability of larger terrorist groups forming also decreases.

 

[54/102 CEF]

There's more obscure stuff than that going on

Google link - start with the 4th item on SOCIAL NETWORKS - if your hair ain't grey it will be when you read this stuff. It can't be penetrated

http://www.google.ca/search?hl=en&q=social+networks+terrorism+sageman&meta=

The guy Marc Sageman briefed on this last year at RMC.

 

[a_majoor]

One of the key concepts in Fourth Generation Warfare (4GW) will be to identify, penetrate and disrupt these sort of social networks.

I would guess that kinetic action isn't always appropriate in this kind of operation, since the largest part of these networks would be passive supporters, followed by active supporters (supply money, safe houses, make cars available etc.) and at the lowest circle of hell are the actual shooters and bombers. I recall an estimate that there were only @ 250 "shooters" in the PIRA at the hight of the "troubles", but they were backed by @ 5000 or so active supporters and a large fraction of the population were passive supporters (i.e. would not interfere with PIRA operations nor report them to the authorities).

Imagine if the passive supporters were converted to "our" side, then we would have tens of thousands of eyes on our side, making movement difficult to impossible for the terrorists. New soldiers will have to be taught techniques to apply this sort of mathematical reasoning in a practical manner, although I have a hard time imagining just how this will be done.

 

[Centurian1985]

A_majoor, this concept isn't new, its been around for quite a while, although with different words and terminology.  The problem is that to do so requires political will that usually alienates and angers a portion of the population.  You cant get anything done until the political leadership recognizes that a legitimate threat to their leadership exists.  LEA and intelligence agencies in countries worldwide have known for years that people of specific ethnic or national backgrounds either openly support insurgent groups in their home countries, or are the target of fund-raising efforts by representatives of those groups.  However, even if you manage to gain political support to investigate and prosecute/deport, there are major barriers in terms of language, ideology, and trust, in trying to penetrate and co-op the communities in question.

 

[a_majoor]

The concept of separating civilians and potential supporters from the Insurgents has indeed been around for a long time. The Romans had a particularly thourough means of doing this ("They create a wilderness, and call it peace"), but in this era we tend to study the French in Indochina, the British in Malaysia and Northern Ireland and the Americans in Viet Nam. All these forces had various meathods to attempt to separate the insurgents from the population, with the British "New Villiages" in Malaysia and the US Marines Civic Action Program (CAP) in Viet Nam being particularly effective.

What is different with this article is the idea that we can move from qualitative effects (it seems to work) to quantitative effects. Knowing the power factors outlined in the beginning article should allow the commander to calculate how many supporters and potential supporters exist in a district. Knowing this, he can then scale his CIMIC, PSYOPS and HUMINT operations to the support base, and scale and time the combat and presence patrols to the expected size of the insurgent cells. Talk about maintaining the aim and economy of effort!

I am far from qualified to discuss the actual methodology of applying this sort of mathematical information to boots on the ground solutions, but there seem to be a lot of possibilities and potential here.

 

[54/102 CEF]

The concept of separating civilians and potential supporters from the Insurgents has indeed been around for a long time. The Romans had a particularly thourough means of doing this ("They create a wilderness, and call it peace"), but in this era we tend to study the French in Indochina, the British in Malaysia and Northern Ireland and the Americans in Viet Nam. All these forces had various meathods to attempt to separate the insurgents from the population, with the British "New Villiages" in Malaysia and the US Marines Civic Action Program (CAP) in Viet Nam being particularly effective.

What is different with this article is the idea that we can move from qualitative effects (it seems to work) to quantitative effects. Knowing the power factors outlined in the beginning article should allow the commander to calculate how many supporters and potential supporters exist in a district. Knowing this, he can then scale his CIMIC, PSYOPS and HUMINT operations to the support base, and scale and time the combat and presence patrols to the expected size of the insurgent cells. Talk about maintaining the aim and economy of effort!

I am far from qualified to discuss the actual methodology of applying this sort of mathematical information to boots on the ground solutions, but there seem to be a lot of possibilities and potential here.

Trendlines are very dangerous measuring sticks - give false idea of accomplishment. Anyone remember the body count?

Jobs for the people, healthcare, education so they move up the ladder from centrally directed dogma open them to links with the rest of the world, opens up their economy and leads back to jobs and a future.

There`s stuff out there that says all the oil producing countries of Islam have a combined GDP of Spain.

Something wrong with this picture?

 

[a_majoor]

Perhaps I am misreading something then, my impression of this and the other article linked in this thread was that these mathematical techniques are not just "trend lines", but can also be predictive. Being able to make accurate predictions and essentially solving the mathematical equations of terrorism would give a 4GW commander a huge advantage in the tactical and operational spheres.

You are very correct to point out that we need to be measuring the proper sets of metrics; not the numbers of dead but the number of captured weapons, the level of economic activities, the number of children in school, the number of contacts our patrols have with civilians, the number of positive, neutral and negative interactions, etc, etc.

 

[54/102 CEF]

The driver of War is politics at all levels.

The IEDs and crack crack crack of small arms means something is not in balance at the political level.

The soldiers have no role there.

And we sure don`t see what the government is doing to fit into the political sphere do we?