Wars follow a power law with a scaling parameter of 2.5

More specifically, the casualty rates of all wars follow a power law, as do the casualty rates within individual wars. Terrorist attacks, likewise, follow the same powerlaw. Different cultures have no impact on these mathematical patterns.

This is useful information, at least in judging the probability of casualties. The intensity of warfare is inversely proportional to its frequency. More importantly, this is the starting point for discovering the mechanisms that lay behind warfare in international relations. War can potentially be viewed as “self-organizing criticality.” It is similar to natural phenomenons like forest fires or economic patterns like urbanization.

Illustrating war as a straight diagonal line on a lognormal graph demands an explanation. In a way, it’s such a rational pattern that it defies rational explanation.

Physicists have been behind many of the most important discoveries outside of physics. This is especially true for the social sciences. Too many social sciences enter with a “humanities” outlook which does not use scientific methods. Physicists are simply smarter than those in English majors and Sociologists.

Not unexpectedly, the study of warfare was propelled by physicists and mathematicians. The invention of Game Theory, thanks to men like von Neumann, helped explain the rational decision-making process behind war. Statistical studies have revealed underlying patterns of warfare. So far, they discover patterns without explanatory theories. This is still a fledgling historical science.

The British physicist Lewis Fry Richardson first discovered the power laws of war casualties There were rare wars of extreme intensity and very frequent wars of low intensity. It was a curious result, and Richardson offered no convincing explanations.

This lead many to speculate about the causes of this statistical pattern. Many of these ideas were eventually shown to be false. Richardson’s description of arms races with Differential Equations was a complete bust.

However, this showed the potential of using scientifically falsifiable hypotheses to explain patterns rather than irrational humanistic explanations.

Roberts and Turcotte revised Richardson’s orginal casualty study. Richardson’s study had a major flaw – it did not fix for population growth. Casualty rates as a percentage of population would be a better measure than casualty numbers.

Using the Small and Singer “Correlates of War” Project data, Roberts and Turcotte found that the casualty rate per 10,000 of the combined populations of the warring nations continued to follow a powerlaw with a scaling parameter of 2.5.
(D. Roberts and D. Turcotte “Fractality and Self-Organized Criticality of Wars” 1998)

This means that wars are scale invariant – they’re fractal.

What do the power laws mean? First, we’re undercounting “wars.” The usual definition of war uses a minimum of 1,000 deaths – the standard used by Correlates of War.

We think of the “hits” but ignore the far more numerous “misses.” Many wars are smothered in the cradle and may not even be recorded. Think about how difficult it is to start an insurgency. You need a political ideology, a network of leaders and followers, equipment, weapons, some amount of popular support aligned against fragile government that is susceptible to attack. Most insurgencies fail before they even kill a single person. Historians may only remember them as a bunch of thugs or troublemakers. But every so often, a thug can cause a major world war.

This shows the problem of measuring failed insurgencies and small wars. The “Pig War” between the United States and the British Empire was an dangerous armed conflict resulting in the death of one pig. This was an acute military standoff which almost resulted in a devastating war. How many pig wars have been fought in history that were never recorded?

We can take the power law and extend it under 1,000 deaths to estimate the number of micro-wars, which greatly outnumber the number of more serious wars. The extreme frequency of small wars and micro-wars must be recorded to get a real understanding of warfare.

This is sometimes compared to forest fires – which follows the same power law. A small spark in a forest usually does not cause a fire. On very rare occasions, it causes a massive forest fire.

Describing frequency of events still does not give us a causal theory with regards to human societies.

This reoccuring mathematical pattern of war gives us new insights. Understanding probability is a major step forward.

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.

Such a powerlaw pattern is produced by self-organizing networks. Networks organize and disperse, and their group strength determines their attack capability:

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.

The model reasonably describes the insurgent organizations I’ve been trying to describe. This seems to accurately predict the damage they cause. If so, then we have a better tool to analyze the dynamics within armed conflicts.

The regularity of conflict in general poses a question Can we explain wars as self-organizing criticality?

Lars-Erik Cederman Modeling the Size of Wars: From Billiard Balls to Sandpiles

Cederman graphed the casualty rates of interstate war using the COW dataset from 1820 to 1997 and Jack Levy’s dataset of Great Powers conflicts from 1495-1965. The results from both datasets show a strong correlation with a power law.

Cederman offers a hypothesis to describe this constant empirical pattern. First, the definition of self-organizing criticality:

Self-organized criticality is the umbrella term that connotes slowly driven threshold systems that exhibit a series of meta-stable equilibria interrupted by disturbances with sizes
scaling as power laws. In this context, thresholds generate non-linearities that allow tension to build up. As the name of the phenomenon indicates, there has to be both an element of self-organization and of criticality. Physicists have known for a long time that, if constantly fine-tuned, complex systems, such as magnets, sometimes reach a critical state between order and chaos.

In other words, this produces the cyclical patterns of war. There is a relatively stable punctuated equilibrium, followed by outbreaks of conflict outside of equilibrium. War is a phase transition.

Here’s the sandpile analogy:

If grains of sand trickle down slowly on the pile, power-law distributed avalanches will be triggered from time to time. This example illustrates the abstract idea of SOC: a steady, linear input generates tensions inside a system that in turn lead to non-linear and delayed output ranging from small events to huge ones.

Roberts and Turcotte also noted this aspect of SOC when they compared warfare to forest fires.

How do wars spread, and why do they spread according to the same pattern as forest fires? IR research shows a “spillover” effect of wars, as warfare in one country is likely to spread to neighbors. There are dampening forces which slow down the spread of war. Logistical limitations and natural barriers (such as oceans or mountains) reduce the spread of war. Technological innovations act as an accelerant.

Robert Gilpin argues that changes in technology and infrastructure act as the catalyst for war. Technology revolutions cause a major shift in economic power and reach – which does not reflect the status quo political power. This brings nations into conflict – the war is a phase transition to a new end state that reflects the new technological and economic reality.

Self-organizing criticality emerges from economic behavior in the international arena. This leads to slow build-up of tensions culminating in conflict. The conflicts emerge from constant interactions of individauls and result in a change in the structure of the system. This build-up of tension is the macroscopic effect of local interactions between agents.

Cederman tests this hypothesis and finds that technological changes and SOC can explain much of the power law pattern in warfare.

Viewing war as a phase transition to a new economic structure may be the best explanation, but is also a very ancient observation. There are definitely other forces at play, but this goes a long way to describing the causes and consequences of warfare.

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