Using mathematical modeling to predict terrorist attacks

A terrorist attack might seem like one of the least predictable of events. Terrorists work in small, isolated cells, often using simple weapons and striking at random. Indeed, the element of unpredictability is part of what makes terrorists so scary – you never know when or where they will strike.

However, new research shows that terror attacks may not be as unpredictable as people think. A paper by Stephen Tench and Hannah Fry, mathematicians at the University College London, and Paul Gill, a security and crime expert, shows that terrorist attacks often follow a general pattern that can be modeled and predicted using math.

Predicting human behavior is obviously a difficult thing to do, and one can’t always extrapolate from past events to predict the future. As one academic discussion of the topic points out, if you made a forecast in 1864 about how many presidents would be assassinated in office based on historical data, the expected number would be zero. But over the next 40 years, four U.S. presidents were killed in office.

Yet when you put individual human acts together and look at the aggregate, they often do follow a pattern that can be represented with math. As Sir Arthur Conan Doyle writes in “The Sign of Four,” the second Sherlock Holmes novel, “. . . while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty.”

The Hawkes process

The mathematical model that Tench and Fry use to look at terrorist attacks is called a “Hawkes process.” The basic idea behind Hawkes processes is that some events don’t occur independently; when a certain event happens, you’re more likely to see other events of the same kind shortly thereafter. As time elapses, however, the probability of a subsequent event occurring gradually fades away and returns to normal.

A mathematician named Alan Hawkes first developed the idea while searching for a mathematical model that would describe the patterns of earthquakes. Earthquake tremors aren’t independent events, either – after an earthquake hits, the area often experiences aftershocks. So Hawkes designed his equations to reflect the greater probability of experiencing a subsequent tremor shortly after the first one.

Since Hawkes developed the model in the 1970s, similar equations have been used to describe all kinds of sequences of related events, including how epidemics travel, how electrical impulses move through the brain, and how emails move through an organization. Recently, Hawkes processes have also been used to predict the locations and timings of burglaries and gang-related violence.

Why gang-related violence follows a Hawkes process is fairly easy to understand. A murder or shooting by one gang often provokes retaliation by another gang. So following the first incident, the probability of a second incident typically goes up.

It’s a little harder to understand why burglaries follow a Hawkes process – i.e., why one burglary would increase the chances of another burglary happening soon after. But, having your house burglarized does increase the chances that thieves will visit again. The burglars now know the location of your valuables and the layout of your house and your neighborhood, meaning your neighbors are more likely to be burglarized in the future, too.

Hawkes processes so accurately describe how trends in crime vary that some security companies and law enforcement bureaus have started to use them in their work. As Fry says, companies like PredPol monitor data on past crimes to model geographic “hotspots” that can be more heavily policed or can become the focus of specific crime-prevention policies.

Predicting terrorist attacks

In their paper, Tench, Fry and Gill apply this same model to terrorism in Northern Ireland. The paper looks at more than 5,000 explosions of improved explosive devices (IEDs) around Northern Ireland during a particularly violent time known as “the Troubles” between 1970 and 1998, when paramilitary groups in the mostly Catholic Northern Ireland fought to secede from Britain and join Ireland. The researchers used the process to analyze when and where one group, the Provisional Irish Republican Army (IRA), launched its terror attacks, how the British Security Forces responded, and how effective those responses were.

IED explosions follow a pattern. After one incident, others follow more quickly. So you have the ordinary chance of the event, but afterward you have a “little kick,” as Fry says, that increases the probability that you’ll have another attack – but then fades away over time. Mathematicians can capture and model these patterns using a Hawkes process equation. The math can reveal patterns in past terrorist activity that weren’t seen before, or be used to test different theories about those patterns, the researchers say. It can also create predictive models, which estimate the probability of future attacks at different times and in different areas.

The researchers say that their analysis shows distinct phases in the conflict between the Irish terrorists and authorities. For example, bombings slowed down as the IRA was infiltrated by British security forces and when more of its members were imprisoned, and bombings increased when the group launched a renewed campaign of violence or tried to use incidents of terrorism as a bargaining tool in negotiations.

One of the most fascinating lessons of the research is on the effects of counterterrorist operations. The paper shows evidence that the death of Catholic civilians, whom the IRA claimed to be representing, would cause the group to increase their IED attacks in retaliation.

That finding echoes previous research that looked at counterterrorism operations by the United States and its coalition partners in Iraq. That paper showed that counterinsurgency operations that were carried out indiscriminately – in other words, attacks that hurt or kill innocent people who were not necessarily insurgents — led to a backlash of terrorist violence. In contrast, counterinsurgency operations that were carried out in a discriminating, targeted way led to a lower level of violence than before.

The paper looks at events in the past, but Tench says the same technique can be used to project future trends. After one terrorist attack, and especially after civilians are killed, the likelihood of subsequent “aftershocks” increases for a specific time period, and authorities need to intervene quickly to avoid a long period of violence. They must also ensure their counterterrorism operations are targeted at the actual insurgents, to avoid provoking the destructive wave of violence that indiscriminate counterterrorism has been shown to do.

Tench says he hopes counterterrorism officials will start using the technique as part of their portfolio. “This application of the Hawkes process is a relatively new idea, so I imagine it might take some time to filter through,” he says.

Baltimore Ravens Offensive Lineman John Urschel Publishes Paper In Math Journal

Some NFL players spend their offseason working out. Others travel around the world. Baltimore Ravens offensive lineman John Urschel has done both while also getting an article published in a math journal.

Urschel, the Ravens’ 2014 fifth-round pick who graduated from Penn State with 4.0 GPA, also happens to be a brilliant mathematician. This week he and several co-authors published a piece titled “A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians” in the Journal of Computational Mathematics. You can read the full piece here:

Here’s the summary of the paper:

“In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.”

When he’s not protecting Joe Flacco, the 23-year-old Urschel enjoys digging into extremely complicated mathematical models.

“I am a mathematical researcher in my spare time, continuing to do research in the areas of numerical linear algebra, multigrid methods, spectral graph theory and machine learning. I’m also an avid chess player, and I have aspirations of eventually being a titled player one day.”

– See more at:

Thanks to Kebmodee for bringing this to the attention of the It’s Interesting community.

Mathematician Safa Moteshari determines collapse of civilization in NASA-funded study


Civilisation is almost inevitably doomed, a Nasa-funded study has found.

Human society is founded on a level of economic and environmental stability which almost certainly cannot be sustained, it said.

The study used simplified models of civilisation designed to experiment with the balance of resources and climate that creates stability – or not – in our world.

These theoretical models – designed to extrapolate from simple principles the future of our industrialised world – ran into almost intractable problems.

Almost any model “closely reflecting the reality of the world today… we find that collapse is difficult to avoid”, the report said.

Mathematician Safa Motesharri begins his report by stating that “the process of rise-and-collapse is actually a recurrent cycle found throughout history” and that this is borne out by maths, as well as historiography.

“The fall of the Roman Empire, and the equally (if not more) advanced Han, Mauryan, and Gupta Empires, as well as so many advanced Mesopotamian Empires, are all testimony to the fact that advanced, sophisticated, complex, and creative civilizations can be both fragile and impermanent.”
His research – funded by Nasa’s Goddard Space Flight Center and published int he Ecological Economics journal – explored the pressures that can lead to a collapse in civilisation.

These criteria include changes in population, climate change and natural disasters. Access to water, agriculture, and energy are also factors.

Motesharri found that problems with each of these is far more damaging when experienced in combination with another. When this occurs the result is often an “economic stratification” and “stretching of resources” which drags at society’s foundations.

Under this highly simplified model, our society appears to be doomed.

In one of his simulations:

“[Ours] appears to be on a sustainable path for quite a long time, but even using an optimal depletion rate and starting with a very small number of Elites, the Elites eventually consume too much, resulting in a famine among Commoners that eventually causes the collapse of society. It is important to note that this Type-L collapse is due to an inequality-induced famine that causes a loss of workers, rather than a collapse of Nature”

He added that elites tend to have a vested interest in sustaining the current model – however doomed – for as long as possible, regardless of the eventual negative outcome:

“While some members of society might raise the alarm that the system is moving towards an impending collapse and therefore advocate structural changes to society in order to avoid it, Elites and their supporters, who opposed making these changes, could point to the long sustainable trajectory ‘so far’ in support of doing nothing.”

There are caveats, of course. The study is a simplified model of society, not a perfect simulation, and it isn’t able to make solid predictions of the future. It’s also worth noting that Motesharri does allow for the possibility that “collapse can be avoided” – though he thinks it will be exceptionally difficult.

Indeed, as the Guardian reports, other studies by the UK Government and KPMG have also warned of a “perfect storm” of energy scarcity and economy fragility coming within a few decades, which lends weight to his conclusion.

2,300 year-old times table hidden in Chinese bamboo strips discovered to be world’s oldest decimal multiplication table

chinese table


From a few fragments out of a collection of 23-century-old bamboo strips, historians have pieced together what they say is the world’s oldest example of a multiplication table in base 10.

Five years ago, Tsinghua University in Beijing received a donation of nearly 2,500 bamboo strips. Muddy, smelly and teeming with mould, the strips probably originated from the illegal excavation of a tomb, and the donor had purchased them at a Hong Kong market. Researchers at Tsinghua carbon-dated the materials to around 305 bc, during the Warring States period before the unification of China.

Each strip was about 7 to 12 millimetres wide and up to half a metre long, and had a vertical line of ancient Chinese calligraphy painted on it in black ink. Historians realized that the bamboo pieces constituted 65 ancient texts and recognized them to be among the most important artefacts from the period.

“The strips were all mixed up because the strings that used to tie each manuscript together to form a scroll had long decayed,” says Li Junming, a historian and palaeographer at Tsinghua. Some pieces were broken, others missing, he adds: to decipher the texts was “like putting together a jigsaw puzzle”.

But “21 bamboo strips stand out from the rest as they contain only numbers, written in the style of ancient Chinese”, says Feng Lisheng, a historian of mathematics at Tsinghua.

Those 21 strips turned out to be a multiplication table, Feng and his colleagues announced in Beijing today during the presentation of the fourth volume of annotated transcriptions of the Tsinghua collection.

When the strips are arranged properly, says Feng, a matrix structure emerges. The top row and the rightmost column contain, arranged from right to left and from top to bottom respectively, the same 19 numbers: 0.5; the integers from 1 to 9; and multiples of 10 from 10 to 90.

As in a modern multiplication table, the entries at the intersection of each row and column in the matrix provide the results of multiplying the corresponding numbers. The table can also help users to multiply any whole or half integer between 0.5 and 99.5. Numbers that are not directly represented, says Feng, first have to be converted into a series of additions. For instance, 22.5 × 35.5 can be broken up into (20 + 2 + 0.5) × (30 + 5 + 0.5). That gives 9 separate multiplications (20 × 30, 20 × 5, 20 × 0.5, 2 × 30, and so on), each of which can be read off the table. The final result can be obtained by adding up the answers. “It’s effectively an ancient calculator,” says Li.

The researchers suspect that officials used the multiplication table to calculate surface area of land, yields of crops and the amounts of taxes owed. “We can even use the matrix to do divisions and square roots,” says Feng. “But we can’t be sure that such complicated tasks were performed at the time.”

“Such an elaborate multiplication matrix is absolutely unique in Chinese history,” says Feng. The oldest previously known Chinese times tables, dating to the Qin Dynasty between 221 and 206 bc, were in the form of a series of short sentences such as “six eights beget forty-eight” and capable of only much simpler multiplications. The ancient Babylonians possessed multiplication tables some 4,000 years ago, but theirs were in a base-60, rather than base-10 (decimal), system. The earliest-known European multiplication table dates back to the Renaissance.

“The discovery is of extraordinary interest,” says Joseph Dauben, a maths historian at City University of New York. “It’s the earliest artefact of a decimal multiplication table in the world.”

It “certainly shows that a highly sophisticated arithmetic had been established for both theoretical and commercial purposes by the Warring States period in ancient China,” he adds. This was just before Qin Shi Huang, China’s first emperor, united the country; he subsequently ordered book burnings and banned private libraries in an attempt to reshape the country’s intellectual tradition.