The Mathematics of War and Terrorism

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The Mathematics of War and Terrorism

Postby Apocatastasis » Tue Aug 09, 2005 6:31 pm

From Statistics of Deadly Quarrels by Brian Hayes:
  • "Figure 3 compares the Poisson distribution with Richardson's data for a group of [outbreaks of] magnitude- 4 wars. The match is very close. Richardson performed a similar analysis of the dates on which wars ended—the "outbreaks of peace"—with the same result. He checked the wars on Quincy Wright's list in the same way and again found good agreement. Thus the data offer no reason to believe that wars are anything other than randomly distributed accidents."
  • "Richardson also examined his data set for evidence of long-term trends in the incidence of war. Although certain patterns catch the eye when the data are plotted chronologically, Richardson concluded that the trends are not clear enough to rule out random fluctuations. "The collection as a whole does not indicate any trend towards more, nor towards fewer, fatal quarrels." He did find some slight hint of "contagion": The presence of an ongoing war may to some extent increase the probability of a new war starting."
  • "Of 94 international wars with just two participants, Richardson found only 12 cases in which the two combatants had no shared boundary, suggesting that war is mostly a neighborhood affair."
  • "What about other causative factors—social, economic, cultural? While compiling his war list, Richardson noted the various items that historians mentioned as possible irritants or pacifying influences, and then he looked for correlations between these factors and belligerence. The results were almost uniformly disappointing. Richardson's own suppositions about the importance of arms races were not confirmed; he found evidence of a preparatory arms race in only 13 out of 315 cases. Richardson was also a proponent of Esperanto, but his hope that a common language would reduce the chance of conflict failed to find support in the data. Economic indicators were equally unhelpful: The statistics ratify neither the idea that war is mainly a struggle between the rich and the poor nor the view that commerce between nations creates bonds that prevent war. The one social factor that does have some detectable correlation with war is religion. In the Richardson data set, nations that differ in religion are more likely to fight than those that share the same religion."
  • "The residuum of all these noncauses of war is mere randomness—the notion that warring nations bang against one another with no more plan or principle than molecules in an overheated gas. In this respect, Richardson's data suggest that wars are like hurricanes or earthquakes: We can't know in advance when or where a specific event will strike, but we do know how many to expect in the long run. We can compute the number of victims; we just can't say who they'll be."


From Scale Invariance in Global Terrorism by A. Clauset and M. Young:
In exploring the distribution of the severity of events in global terrorism, we have found a surprising and robust feature: scale invariance. Traditional analyses of terrorism have typically viewed catastrophic events such as the 1995 truck bombing of the American embassy in Nairobi, Kenya, which killed or injured more than 5 200, as outliers. However, the property of scale invariance suggests that these are instead a part of a statistically significant global pattern in terrorism. Further, we find little reason to believe that the appearance of power laws in the distribution of the severity of an event is the result of either reporting bias or changes in database management. This suggests that the power law distribution, with alpha = 2, is an inherent feature of terrorism and counter-terrorism. Indeed, the severity distribution itself has changed very little over the past 37 years of recorded events (Fig. 2), in spite of a dramatic increase in the frequency of events. This small growth in the breadth of the severity distribution may be the result of technological changes, such as the power and availability of cheap explosives and firearms.

Surprisingly, the scale-invariance result extends beyond the total collection of events. When we examine the distributions for major industrialized nations versus the rest of the world, we find that heavy tails are present in both (Fig. 1c), but with substantially different exponents: G7 = 1.71 ± 0.03 versus nonG7 = 2.5 ± 0.1. That is, while events occur much less frequently in major industrialized nations, when they do, they are much more severe (on average) than events outside those nations. Additionally, when events are partitioned by weapon type, statistically significant power laws persist (Fig. 3, Table II) and show that any roughness in the scaling of the aggregate distributions (e.g., Fig. 1a) is derived from the composition weapon-specific power laws with distinct scaling parameters and ranges. It also illustrates that there is something unique about explosives, which causes the shallow scaling of the lower tail for the injuries severity distribution.


From From old wars to new wars and global terrorism by Johnson, Spagat and some other fellows:
Even before 9/11 there were claims that the nature of war had changed fundamentally. The 9/11 attacks created an urgent need to understand contemporary wars and their relationship to older conventional and terrorist wars, both of which exhibit remarkable regularities. The frequency-intensity distribution of fatalities in "old wars", 1816-1980, is a power-law with exponent 1.80. Global terrorist attacks, 1968-present, also follow a power-law with exponent 1.71 for G7 countries and 2.5 for non-G7 countries. Here we analyze two ongoing, high-profile wars on opposite sides of the globe - Colombia and Iraq. Our analysis uses our own unique dataset for killings and injuries in Colombia, plus publicly available data for civilians killed in Iraq. We show strong evidence for power-law behavior within each war. Despite substantial differences in contexts and data coverage, the power-law coefficients for both wars are tending toward 2.5, which is a value characteristic of non-G7 terrorism as opposed to old wars. We propose a plausible yet analytically-solvable model of modern insurgent warfare, which can explain these observations.


Power laws, yet again making the world a more interesting place and making ideologies right and left look silly, silly, silly.
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Postby Sturm » Wed Aug 10, 2005 6:22 pm

But are we talking probabilities of an occurence/trend vs historical repetiviness or choas/quantam math.


great read*(still reading, first link & others), felt my brain grow...*shudders lol. :wink:

Wonder if i could put this in my sig...j/k
Even if Richardson's limited data were all we had to go on, one clear policy imperative emerges: At all costs avoid the clash of the titans. However painful a series of brushfire wars may seem to the participants, it is the great global conflagrations that threaten us most.

As noted above, the two magnitude-7 wars of the 20th century were responsible for three-fifths of all the deaths that Richardson recorded. We now have it in our power to have a magnitude-8 or -9 war. In the aftermath of such an event, no one would say that war is demographically irrelevant. After a war of magnitude 9.8, no one would say anything at all.

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Postby Radix » Wed Aug 10, 2005 6:59 pm

Even if Richardson's limited data were all we had to go on, one clear policy imperative emerges: At all costs avoid the clash of the titans


Strange....what I took away from it is that such wars are unavoidable.

I'm no mathematician, but it seemed like, the scale being defined by the author, the whole article might just be an excercise in tautology obfuscation. "Hey look, there's a nice pretty pattern when you put it on a logarithmic scale (nevermind the fact that I'm intentionally defining the breaks between the levels so it always works that way)!"

A longer study woould have been better. The down side is that history hasn't always been recorded very well, even during the years they did look at.

I'd be interested what they considered "war" to begin with. International, uniformed conflicts are the easiest to identify, but by no means are they the only battles that define human conflict. I can't help but think that either the study is severely limited (by only counting the easily identifyable wars), or severely incomplete (I doubt every small conflict over the globe over the last nearly 200 years was counted).

Those, however, are complaints best left to the authors' peers, not some schmuck with a calculator. Interesting reads anyway.
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Postby Sturm » Wed Aug 10, 2005 7:24 pm

Me ethier ..i see an algorithmic symbol..my brain goes blank..lol

I took , that even with the infrequency of mag 7+ war , the lower mag 4-6's, because of frequency etheir evened out or were greater in the long run in the terms of cost(lives/Weapons&Ammo/$$$..etc)

I wonder what the numbers would show if a study was done from the first 1000 years to present.
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Postby Apocatastasis » Wed Aug 10, 2005 9:11 pm

But are we talking probabilities of an occurence/trend vs historical repetiviness or choas/quantam math.


Probability distribution of the onset/end of wars. If there were "historical repetitiveness" factors, it would have been noticed in the statistical search for correlations. The modelling is unrelated to chaotic systems, it is not trying to model the behaviour of a single war but the aggregate statistical look of wars as abstract events described by some parameters, I think.

Strange....what I took away from it is that such wars are unavoidable.


Not at all. High scale wars are less probable to happen than smaller scale ones, and neither kind is "unavoidable". Think of this: is it "unavoidable" that you will have an earthquake in the Pacific Rim during the next 6 months? No, it's just very probable, in the sense of a formal probability event.

I'm no mathematician, but it seemed like, the scale being defined by the author, the whole article might just be an excercise in tautology obfuscation.


Again, not at all. The scale is chosen to make the unavoidable biases and ignorance about exact fatality count of little significance for the search of statistical patterns. The pattern that emerges, as is the repeatedly verified and accepted find by now, is that conflict intensity measured in term of fatalities follows a power law distribution ("scale invariance" is another way to put it, basically meaning the same here). The pattern might not have existed at all (for instance, they might have found no regularity, just a uniform random distribution), a normal (bell shaped) one, or even more funny ones. Knowing there is a pattern of distribution and which one it is is a BIG deal. The predictive results are much different if you find the discovered pattern is some other distribution. Also, knowing which distribution it is can give you important clues to understand the event you are measuring in the aggregate. In this case, for instance: power laws are well known as the observed behaviour of complex adaptive systems (for which chaos maths and network theory modelling has been proved fundamental). This hints that an individual war might be similar to a complex adaptive system, which is a good thing to know and (for me, at least) would explain stuff I've observed (for instance, why optimization systems used in business resource allocation fail miserably to predict war supplies allocations). But now we are just entering fuzzy speculation realm so I'll leave it there.

I'd be interested what they considered "war" to begin with.


I think the first article mentions Richardson's criteria, he seems to have included all manner of low scale violent incidents too and the pattern holds for them also. For what I understand, bigger data sets have been collected and processed and still no fault can be found in Richardson's basic conclusions. I think they are concentrating on refining them by now, mainly, and checking for new factors that weren't of significance by the time he did the original work (one I could think off the top of my head: does the number of international organizations a country belongs to correlates in any way with the number of wars participated in?).

The two works on terrorism are more controversial but for what I've been following the consensus is moving towards acepting that terror fatalities also follow a power law distribution (2nd work) and so can be studied as real wars with different parameters (3rd work, which seems to present an interesting model of terror group behaviour using maths that look to me related to network theory (I'm wayyy rusty on that area)).

Hope that helped, sorry if I was too dry up there in the original post. :P

[Edited umpteen times because I type with my forehead before my coffee.]
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