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Math vs. Mayhem

In the 1970s, mathematician Alan Hawkes was searching for a model that would describe the patterns of earthquakes and the resulting tremors that occur in the wake of the quake.

Most earthquakes generate aftershocks, because they increase the geological tension in the region in which they occur. Hawkes wanted to develop a mathematical model that could measure how much more probable aftershocks become after an earthquake, and how long that increased probability holds sway. 

He succeeded. His work, later called the Hawkes Process (HP), models a sequence of "arrivals," or related events that occur in a series. The defining characteristic of the HP is that it "self-excites"—each arrival increases the rate of future arrivals for a certain period of time. Then, as time elapses, the probability of a subsequent event gradually fades away and returns to normal.

As time passed, scholars discovered that the HP model could be applied to other events that also self-excite. A fight between rival gangs, for instance, often ignites a series of violent episodes of revenge and retaliation. Similarly, the collapse of a Wall Street investment bank can send shock waves through world markets that may cause a series of financial companies to collapse.

"HPs are fundamentally fascinating models of reality," wrote researchers Patrick J. Laub, Thomas Taimre, and Philip K. Pollett in a 2015 scholarly paper, Hawkes Processes. "The HP is structured around the premise that the history [of an event] matters, which partly explains why they appear in such a broad range of applications."

Indeed, in the last few decades, HPs have been applied to a range of situations, including cancer tumor recurrence, wildfire hazards, and social interactions. Recently, the HP has been particularly popular in the area of quantitative finance, such as in the analysis of financial market data.

Several years ago, HPs were applied to crime prevention and predictive policing. A few police departments, including those in Santa Cruz and Los Angeles, began using computer software containing HPs to anticipate where and when crimes might occur.

Burglary, these departments found, is often a self-exciting activity—when a house is burglarized, it increases the chances that thieves will visit again. This is because the burglars now know the layout of the house and where the valuables are located. Moreover, the burglars also know the neighborhood, so neighbors are also more likely to be burglarized in the future.

With HP-enabled programs, data on past crimes can be used to model geographic crime hotspots that can be more heavily policed or can become a focal point for special crime-prevention policies.

Now, HPs are being used in the area of terrorism. In one recent study, Spatio-Temporal Patterns of IED Usage by the Provisional Irish Republican Army, researchers Stephen Tench, Hannah Fry, and Paul Gill argue that the HP can be especially useful in modeling the back-and-forth interactions between insurgent attacks by militant groups and the counterinsurgency operations of those fighting the militant groups.

In arguing for this usefulness, the researchers cite another study in the field as evidence. In this study, Self-Exciting Point Process Models of Civilian Deaths in Iraq by Erik Lewis, George Mohler, P. Jeffrey Brantingham, and Andrea Bertozzi, the authors used an HP model to study the more than 100,000 violent civilian deaths that occurred in Iraq after the U.S.-led invasion, Operation Iraqi Freedom, began on March 20, 2003.

The authors examined temporal patterns of violent deaths for four different regions in Iraq: Karkh, Mosul, Fallujah, and Najaf. In three of the four regions, the researchers found that each initiating event, or attack, generated slightly more than one self-excited "offspring event." For three of the regions, the average interval between initiating and offspring event was brief, ranging from eight to 17 days.

"The briefness of this interval could correspond to the amount of time a hostile actor needs to prepare for another attack," the authors wrote. In the fourth region, Najaf, every two initiating events generated approximately one self-excited offspring.

Knowing these time frames can assist in developing quick interventions that could help lower the problem of violent deaths in Iraq, the authors found.

After discussing this study, Tench, Fry, and Gill discussed their own findings. The authors looked at more than 5,000 explosions of improvised explosive devices (IEDs) around Northern Ireland during a particularly violent time known as "the Troubles" between 1970 and 1998, when paramilitary groups in Northern Ireland fought to secede from Britain and join Ireland.

The researchers used the HP to analyze how the Provisional Irish Republican Army (PIRA) launched its terror attacks with IEDs, how the British security forces responded, and how effective the responses were.

IED attacks often self-excite—after one incident, others follow more quickly. And so, just as HP-based models are effective in measuring earthquake-and-aftershock patterns, they fit the terror attack-and-response process too. "On a mathematical level, they are quite close," Tench told Security Management in a recent interview.

In terms of time frame, the authors focused on the PIRA's active period of 1969-98. This period has been divided into five distinct phases by previous researchers who used a qualitative approach, based in political science and sociology, to analyze PIRA operations. For example, in phase one (1969-76), the PIRA organized itself in military arrangements of brigades, battalions, and companies. In phase two (1977-1980), the PIRA fractured into smaller groups, using a more informal cell-based approach to operations.

But Tench, Fry, and Gill, using a mathematical HP-based approach, found that this five-phase model could be refined further, as transitions between the phases were more fluid than originally described.

"The (phase) boundaries did not work as well as they thought they worked," Tench says. This presents an opportunity for qualitative researchers to further refine the five-phase model, using the new mathematical findings, and reach a better understanding of the PIRA's operations. "That would be an interesting development on the qualitative side," Tench adds.

The researchers' analysis also yielded insights into the effects of counterterrorist operations. The authors found that that the death of Irish Catholic civilians, for example, resulted in more frequent retaliatory IED attacks by the PIRA.

Buoyed by the results of the current study, Tench said that he and his colleagues will be conducting more HP-based analysis in the near future on other terrorism attack-and-response situations in places like the Middle East and Africa. For example, his team has been looking into more recent events in Somalia involving the militant group Al Shabaab.

"Taking it forward, we'll be trying to take it into a modern context," Tench says. "The Northern Ireland case is kind of a testing ground." ​