Log ind

The Mathematics of War: Quantifying Insurgencies


Daniel Brooks, M.Sc student, Niels Bohr Institute and the Royal Danish Defense College.


The aim of this article is to call attention to a mathematical model that treats modern insurgency as an ecological system and how this analysis may provide commanders with valuable feedback on the effect of their strategies on insurgent networks. This model – published in the internationally renowned magazine Nature – describes insurgency as an ecology of dynamically evolving, self-organized groups following common decision-making processes.’1 The possible avenues of applying this model to provincial or district-scale battlespace in order to raise situational awareness and understanding is the subject of this article, and the research project I am conducting in this area is introduced.

Qualitative and Quantitative Approaches to Intelligence Analysis

In intelligence analysis there are essentially two favored approaches. One is expert assessment, where subject matter experts rely on extensive experience and knowledge of their fields and from this draw conclusions based on available knowledge and their own educated guesses. As is clear to analysts, the key difficulty to this approach is the unavoidable reliance on a mental model. ‘Mental model’ is a term that covers the body of experience and knowledge that offers the analyst a certain perspective or viewpoint. Mental models serve as filters/paradigms, telling the observer what is important and how to interpret data.2 An alternative approach to intelligence analysis is so-called Structured Analytical Techniques (SAT) which is a toolbox of logic techniques that lends some degree of quantifiable value to a difficult analysis that often must be based on multiple assumptions of varying reliability that place even experts on shaky ground. SATs give experts the tools necessary to collaborate in a structured manner and root out common biases in their analysis. A third approach that has fallen out of favor for a number of sound reasons is actual scientific inquiry based on mathematical models. With modern tools of intelligence gathering and information management however it has become feasible to start applying quantitative models based solely on data without going through the filters and paradigms of analysis. The attraction to this approach is that it does not fall afoul of a number of cognitive biases that limit the expert in the mental model. The major problem with the comprehensive, quantitative approach however is the limited data set. Because the simple fact is that it is impossible to know all the ends. A complete picture of a given battlespace will never be available to the intelligence analyst, not only because insurgent forces do their very best to hide their activities and organization, but also because the insurgent network cannot in truth be called a singular organizational network.

Systems and Intelligence

In the target-centric approach to intelligence analysis, all targets are considered systems, consisting of structure, function and process. Furthermore, in irregular war, almost all targets may be considered complex systems, because they are in a constant state of dynamic evolution and thus behave in non-linear fashion.3 Non-linearity is an important concept in science. Linear systems are of interest to natural science because it is possible to solve them analytically: the variables examined can be separated and evaluated independently. In non-linear systems however, this is not possible as the system as an observable phenomenon will break down once we begin examining its components separately; out of context of related system components. A relevant model to examine to see the difference between non-linear and linear systems is the organization of military and insurgent organizations. One can easily construct a hierarchy of command and structure when looking at a military unit, whereas constructing a hierarchy of an insurgent organization is at best very difficult with numerous cross-links between subordinates. Insurgent organizations are thus usually referred to as networks, because of this highly interlinked nature. However, as is known to anyone who has worked in the intelligence business of irregular war, reality is even more complicated than one might expect and in fact often exceeds the analytical capacity of the human mind. Insurgencies cannot simply be described as networks of fighters, but must rather be seen in the context of an extremely complex ecosystem of causes and effects. So what is the use of attempting to do quantitative science on a vastly complex system when most of the data is essentially unavailable?

Power Laws

The answer lies in what kinds of observables we do have access to. A lot can for example be said of the complex inner workings of our Earth from observing the patterns of earthquakes, or the flow of water in an ocean by watching the surface currents. When we examine systems for observable attributes, we are almost always using some kind of statistical trick to map reality into graspable numbers. For example, evaluating the performance of soldiers in a military unit usually involves doing statistical averages on scores of personal abilities. Attributes like grades usually follow a bell curve, also known as the normal distribution. Normal distributions are entirely defined by their mean and variance, which gives very convenient, simple numbers to report. However, normal distributions are usually found when events are independent of each other. Once the interdependence of events become important, the normal distribution fails, because feedback loops tend to amplify small initial events. This can be observed in a statistical relationship called a power law, or Pareto distribution. When the frequency of an event relies on the power of some attribute of that event, such a size, it is said to follow a power law. Many complex, non-linear systems, where events are highly interdependent, exhibit this particular behavior. Examples include earthquakes, stock market fluctuations, traffic to websites, popularity of authors, and as it turns out, kinetic events in irregular wars. Power laws do not have a clearly defined average, so they do not lend immediate, simple numbers to those looking for them. Simple numbers, however, are likely to represent a static image of the systems whose attributes they represent, and there is often a tendency to look for the static when one should be examining a dynamic, and this is where power laws have their greatest use.4 It has been established in prior research by Gourley et al that attack distribution in an irregular war, when considering intensity versus frequency, follows a power law, f(x)=ax^-k and also that the stability and hence the continuation of the war is correlated with the slope of this power law. The conflict can be expected to continue when k~2.5 whereas it can be expected to come to an end when k>=3 or k<=2.

Skærmbillede 2020-01-08 kl. 14.23.11.png

Dynamic Battlespace Analysis

While an irregular conflict is a beast of astounding complexity, the discovery that something as central as the distribution of attacks might provide insight into the ebb and flow of related dynamics is of interest to both military strategists as well as members of the public wanting to inform themselves on the course of a conflict their nation is invested in without the embellishment of facts that automatically comes with the territory. An irregular conflict is by its nature a combination of kinetic actions and politics, where the battlefield is located not just on the physical terrain but just as much in the subjective, cognitive space of people's opinions on their situation. The main objective of most of these conflicts is the creation or dissolution of stability, and because stability is a subjective notion in the minds of civilians in both the arena of conflict as well as the political arena that supports the military efforts, information is almost always filtered through a number of directed interests, rendering most reports on the conflict editorialized beyond reasonable accuracy. Reliable, objective information has become a rare commodity indeed, and an objective tool for assessing the actual state of a conflict based on publicly available data such as reports of kinetic activity is worth pursuing and developing in the interest of a well-informed public. Meanwhile, military strategists who face the complexity of the network-based battlespace on a daily basis might find a useful tool in a model that provides information on the time-evolution of the network that produces kinetic activities. The system of causes and effects in a counterinsurgency operations battlespace like Afghanistan is a daunting maze of effects feeding into each other, each of which usually also represents a network in themselves. Military intelligence is tasked with providing a clear picture of each of these factors, the primary of which is human terrain mapping, with a focus on red force (enemy) identification and inquiry. This process usually uses a detail-oriented approach, gathering information on the individuals and their interactions from a number of source types such as IMINT (Imagery Intelligence), concerned with the analysis of images from operatives or orbital satellites, HUMINT (Human Intelligence), concerned with person-to-person intelligence gathering and SIGINT (Signals Intelligence) usually concerned with eavesdropping on the various means of electronic communications. These parts of the intelligence apparatus are all bottom-up approaches where as many details as possible are fused into a broad picture. This is considered a robust, logical system which provides the most accurate and unbiased version of reality that commanders in the field can act upon. But no matter how robust and how many resources are put into the gathering of intelligence in this manner the drawback is of course that one cannot know everything. Information needs to be corroborated, often by multiple intelligence types, and some things just always fall under the radar either because the individuals in question are very careful to keep their activities hidden, that they operate in a grey area of the complex network of causeand-effect that intelligence, having limited resources, isn't concerned enough with, or that their signal just doesn't make big enough waves to rise above the noise level and figure on the intelligence gathering apparatus' channels. A top-down approach can therefore prove itself useful to military strategists wanting to provide commanders with a broad picture of the situation, based on the best available data, but taking into account the holes in the map of details. Some forms of intelligence gathering work like this, notable among which is MASINT (Measurement and Signature Intelligence) which employs remote sensing techniques to map out things like general heat activity in an area, or compares highresolution images to assess whether the ground has been disturbed, indicating IED emplacement Top-down approaches like these can be difficult for the analyst to put into practical intelligence, especially for lower levels of command such as Battle Group, and the power law model approach dealt with in this research project is likely to have its limitations. The power law model does not provide a 'war forecast' and can't tell a commander where the next attack is going to occur. It can, however, give a broad sense of how a newly employed strategy is affecting the battlespace and the insurgent network with time. Though, if this model is to be useful to anything below the very top level of control in the conflict, it all depends on whether the model exhibits a property known as scale invariance.

Scale Invariance

A central aspect of power law distributions is their scale invariance. A system producing a power law behavior ought to be able to exhibit the same properties when a subset of this system is considered. What this means for the concrete project at hand is that the power law behavior of the Afghanistan war exhibited on a national scale should also exist when looking at the conflict on the provincial level. The obvious choice here is of course Helmand Province where the Danish forces are deployed and the author has direct intelligence experience as well as access to raw intelligence. The challenge with provincialor district-level attack distribution is the resolution of data. Because Gourley et al did not have access to military intelligence they had to rely on newspaper clippings to provide them with data on attacks. The downside to this is obviously that only attacks that cause casualties are reported in the media. On the contrary, the advantage to a data set such as this is that there is a very reliable, linear number that can be used as an intensity measure: number of deaths caused. When faced with a much higher resolution of data presented by actual intelligence, it becomes a challenge to create a linear scale for measurement of intensity, when no mortal casualties are involved.

The Research Project

The detailed aims of this research project is to:

 Test the power law model of human insurgency against a higher resolution dataset supplied by military intelligence.

 Test the scale-freedom of the model of human insurgency when applied to a smaller subset of the Afghanistan conflict.

 Develop a numerical simulation of the insurgency network in Helmand, based on both the network database as well as intelligence reports on the organizational structure of the insurgency. Subsequently test if it can produce the same power law behavior.

 Discuss practical implications and applications of these models.

Scale-free networks

The 2.5 slope is interesting to consider as it invites a relationship with a number of networks found elsewhere in nature. For example, an auto-catalytic network in biology, the self-organized system thought responsible for the origins of life, exhibits what appears to be a very relevant property: its degree distribution follows a power law and is stable and functioning only when this power law has a slope between 2 and 3. The similarity is striking, and it would therefore be of value to try out some of the models of scale-freedom on the insurgency ecosystem. Another important aspect in assessing the scale-freedom exhibited by insurgency systems is to find out how small a scale, in time and space and group size the model still works. This would be of value because it can represent a boundary condition on the usefulness of the model to provide feedback on strategic decisions of commanders and strategists. A test of the model, both numerically and on real data, on smaller and larger scales, is therefore called for: Theatre-level, provincial-level and district-level.

Directed Network-based model of insurgency

While the ecosystem model of insurgencies is, despite the complexity of the system it models quite mathematically simple and reproduces the observed power law behavior when run through numerical simulations, it would be interesting to see if a directednetwork model exhibits the same. To do this, I aim to design a new model from scratch, considering the train of resources from sources such as out-of-country support lines through intermediaries such as IED workshops and into the hands of the end users who bring the actual attack to fruition. This model would include a grouping and fragmentation element, but instead of an addition of work as communication or collaboration happens, the groups themselves would form a network on which work that goes into an attack travels from sources to sinks, gaining momentum before it goes into the final product, the attack. Whether this is a useful change to the model remains to be seen, but intelligence experience suggests that the insurgency effort is highly reliant on this train of resources where many insurgent groupings serve in supporting roles to commanders in the field. For example, an IED maker is a very high value target for coalition forces in Afghanistan because though he may never conduct attacks himself directly, he represents a concentration of effort by the insurgency in the areas of training, materials support and logistics that is of central importance to its effectiveness.

Processing SALTAs and 9-liners

The most consistent source of data in the Afghanistan theatre is the JCHAT message board, where all incident reports are logged as they happen. A SALTA is a first-response attack report filed by the Tactical Operations Center (TOC) the given unit under attack belongs to. A 9-liner is a request for medical assistance in the field. These are valuable sources of data, not least because they are presented in a very precise format that may be mined for data easily. Nine-liners and SALTAs contain exact positions in time and space, as well as count of casualties, deaths as well as injuries. However, this may not be enough. A kinetic incident is a messy thing where minute chaotic details become the difference between life and death for the parties involved. While on the large scale a casualty count may be sufficient, when dealing with a higher resolution data set, it becomes tricky to factor in all the elements of an attack merely from the unlucky statistics of death and injury. It may become necessary to go through follow-up reports to gain a clearer picture of how much effort and work has gone into a kinetic incident, as this is ultimately the property which is important.

Gathering intelligence

It is possible to base a model on real intelligence gathered on the observed structure of insurgent networks in the field. These observations are by their nature synthesized from a number of different sources, almost all of which are classified. This poses some challenges to the gathering of useful data, since this project document itself will be unclassified. The anonymization of classified data is tricky because it isn't from the outset clear which parts are essential to building a functional model. Nevertheless, it will be possible to append a classified section to the project which will serve to give strategists a clearer contextualized picture of the usefulness of these models to feedback and decision-making. Another issue with gathering useful data is that most available intelligence has been pre-analysed and filtered before being entered into the database. Added to this issue is that the analysts are exchanged in regular rotation, and as they gain experience and are replaced, their methods and focus will shift. A solid data source would therefore need to be unfiltered, and the most promising data source for this is electronic signal intercepts. While the value to an all-source intelligence cell of this raw data is problematic before it has been analysed, the aims of this research project is as indicated earlier, not to gain a complete knowledge of the intelligence situation, but rather to analyse the dynamic evolution of it.

Concluding Remarks

As the network-centric battlespace gains credit as a paradigm in intelligence analysis, it becomes useful to apply the tools of complex systems analysis to “the box”. In the extremely complicated realm of modern irregular warfare, it is impossible to know all the factors. However, in the same fashion as it is possible to gain knowledge about the internal dynamics of an ocean by observing the waves on its surface, so it may be possible to say something about the complicated and chaotic ecology of cause and effect at work in insurgent activities from observation of their surface activities. While the author holds no illusions about being able to predict the outcome of the Afghanistan conflict, military strategists are urged not to disregard the potential power of quantitative tools of intelligence analysis to provide situational awareness and understanding of an objective nature.


1 Juan Camilo Bohorquez, Sean Gourley et al, Common ecology quantifies human insurgency, Nature 462, 911-914 (17 December 2009)

2 Heuer, Pherson, Structured Analytic Techniques for Intelligence Analysis, CQ Press, 2011, p. 5

3 Robert M Clark, Intelligence Analysis: A Target-Centric Approach, Chapters 1-2 CQ Press, 2010

4 John Hagel, “The Power of Power Laws”, http://edgeperspectives.typepad.com/edge_perspectives/2007/05/the_power_of_po.html