The Math Equation That Explains David and Goliath

From babies' tantrums to labor strikes to guerrilla wars to global terrorism, there may be one simple math equation, a power law, that benchmarks them all. Better yet, it may allow us to predict these confrontations' future.
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From babies' tantrums to labor strikes to guerrilla wars to global terrorism, there may be one simple math equation, a power law, that benchmarks them all. Better yet, it may allow us to predict these confrontations' future.


Brazil’s protests started in 2012 with demonstrations about the cost of public transit that morphed this year over the Southern Hemisphere’s winter as concerns over public spending on next year’s World Cup and 2016’s Summer Olympics spurred larger gatherings in more places. The protests have dwindled, but it’s anyone guess if the so-called Brazilian Autumn will flare up again in 2014.

What if it wasn’t necessary to guess?

A gaggle of academics suggest the timing and size of protests—and of a basket of other conflicts, ranging from crying babies to global terrorism—can be predicted using a mathematical equation.

And if it predicts the trajectory of confrontation, can intervention be far behind?

A new paper in Nature Scientific Reports looked at a number of publicly available datasets on asymmetrical—David versus Goliath—conflicts, and notes that both their severity and timing followed a similar statistical pattern. The more the researchers looked, adding on studies of similar asymmetrical incidents like sexual assaults on women, online trading attacks on a company’s share price, or cyberhacking into a nation’s infrastructure, the more times they found the equation fit the distribution.

“This suggests it’s an innate property of the two-sided asymmetric confrontation,” said lead author Neil F. Johnson, talking briefly like the physicist that he is. “We think that when you got a group of humans who are self-organized on the fly and coordinating themselves to oppose some bigger power, this is exactly what you’re going to get. It’s the way that they engage themselves, small and adaptive versus big and sluggish.”

A gaggle of academics suggest the timing and size of protests—and of a basket of other conflicts, ranging from crying babies to global terrorism—can be predicted using a mathematical equation.

The “simple math law” Johnson gets is known as a power law. As one of his 16 co-authors, economist Michael Spagat of London’s Royal Holloway College, explained: “The number of events of size X is going to be some constant, times X, raised to a certain power—hence the power law. The power turns out be right around negative 2.5.” On a chart, the various incidents lie on or very near a nice, straight line, and it’s in extending that line beyond what’s already logged that we can arrive at a prediction. Different assumptions—say about how likely guerrillas are to stay within a group or dissolve their militias—can change the specific placement of points, but the power law still tends to establish an accurate pattern.

Johnson’s underlying research (but not necessarily that of all 16 of the co-authors) is being paid for by the Intelligence Advanced Research Projects Activity, or IARPA, an arm of the American spy network that is going all out to forecast the future. He’s looking specifically at using only open-source material—think Facebook postings—to generate predictions about civil unrest in Latin America.

“In order to be able to do that, you have to able to identify some sort of pattern—is there some kind of pattern of behavior in the ways that civil unrest evolves,” he explained. “We have to produce warnings about events that are actually in the future ... is there any reason to think that there will be a generic way in which a protest will begin to pick away at an incumbent government.” (His is one three groups in this particular challenge from IARPA.)

If this sounds faintly science-fiction-y, it is. Isaac Asimov’s Foundation series of stories centered on a discipline—psychohistory—that combined math and social science to divine mankind’s path. The non-fictional new discipline of cliodynamics is taking a similar if opposite tack, reverse engineering events to find the equations behind history. But even before this, a hundred years ago Lewis Fry Richardson first postulated there might be power laws in human violence. Building on his work and others’, the academy took the possibility of power laws to its heart—perhaps a little too closely. “There was quite a fad—I’d say we’re past that now—for finding power laws everywhere,” Spagat said. He notes the work of academics like Mark Newman, Cosma Shalizi, and Aaron Clauset in setting down the statistical techniques that turned a parlor game into a predictive tool. “They raised the bar quite a bit for what you have to go through before you can say for certain there’s a power law.”

Johnson and Spagat’s own walk through this particular gate started about a decade ago in Colombia, where a nasty insurgency was producing body counts—and data about the series of individual incidents. Johnson attended a lecture by one of Spagat’s students on the distribution of casualties from attacks. He saw parallels with his own interests.

Johnson runs a research group at the University of Miami that looks at “complexity,” and while he started his career looking at what happens with one set of particles embedded in a larger set, he’s branched out to people. "They’re exactly analogs of this except the particles don’t adapt," he said. "They either took over or it didn’t take over. It wasn’t like that it would try to take over and the other side would fight back, and then the first side would adapt itself. That’s much more interesting in my opinion."

In that Colombian lecture, Johnson’s interest was piqued. “Distributions are normally bell shaped, so when something that’s not bell shaped comes along, you ask, ‘Is there some sort of order in the madness?’” he said. In the case of Colombia, and as they folded in individual event data from the Iraq War, order did emerge. “Not only were they not like bell curves, they were like power laws. This set us on a long, long road to see what was going on.” (Here’s an earlier study derived from this work looking at the “ecology” of conflict.)

"We looked at the old wars that Richardson was looking at a hundred years ago," Johnson said. "The funny thing was at the time the wars around him weren’t the right type for power laws. What we found was that these old wars ... where armies of equal ability fight it out, the casualties in that kind of war were really kind of like a random system. When we looked at all the kinds of war that happened in the mean time, wars across Africa, Afghanistan, and also global terrorism, those didn’t fit the bell curve."

The researchers kept piling on data. Colombia. Iraq. Afghanistan. A compendium of African war incidents from the University of Uppsala. Northern Ireland. All manner of places where little forces “pick away” at the bigger state, where Davids fragment and then coalesce over and over.

“It didn’t seem to matter if you looked at it on the individual, scale of a country, the whole country, or the whole continent,” Johnson said. “There was something going on in those sorts of wars that wasn’t going on it conventional sorts of war.

“It turn out that it has nothing to do with the violence,” he concluded, “It has to do that when you’ve got two sides ... and one side is thought of as working on the fly. Because it’s working on the fly it’s adaptive, it’s quick to take up opportunities. It’s fighting some sort of Goliath that might be the incumbent state that has a formal way of replying and a formal way of behaving. That’s when we get these power laws.”

“It doesn’t always work perfectly,” Spagat cautioned. “But it works really well in a variety of settings, a wide variety of settings.” (One outlier is the civil war in Angola. “Angola sits way off—to this day we don’t know why,” Johnson said, speculating that it may be the way that particular data was recorded.)

“But it fits for the terrorist organizations, it fits for the opposition to the Coalition in Iraq, it fits for the FARC down in Colombia, it fits for all these types of wars that are two sides lining up in unequal ways,” Johnson said.

Initially the academics were looking solely at the size of events. Then they turned their attention to timing. “This is where things get really curious,” Johnson said. “We found that way in which timing escalates—or de-escalates—across different examples of the same type of confrontations ... the way in which they escalate falls along this same straight line.”

More researchers turned up with more data sets, and the relationship held up. Hacking. Sexual assaults. Strikes. And babies crying. In a laboratory setting, parents were told to not respond to their baby’s crying, and the pattern of each baby’s response was noted—crying, not crying, not crying, crying, wailing, etc. Likening the baby to an attacker, Johnson described them as “a small entity, very agile, very quick to adapt in interacting with a parent.” But among different babies and different parents, the timing and scale of the kids’ outbursts fell along the same power law curve.

Johnson said he’s seen nothing outside the human realm (so far) that replicates this power law. He notes that automated attacks on a company’s share price isn’t directly organic, but those computers in turn were programmed by flesh-and-blood coders. (He’s currently interested in the commercial side of the equation, too, when a small company begins to pick off successors against a larger one, or a product begins to make an inroad against another product, a la VHS vs. Betamax.)

But it’s all academic until the power law proves so robust that Goliath can intervene and alter a pre-ordained event. Johnson said he thinks that will be possible, and it’s a good thing, too, because most of the future wars are expected to be asymmetrical. (Or it could be a bad thing, if you were supporting Davids in Egyptian or Ukrainian protests and not their Goliath incumbent governments.) The key, he suggested, will be to intervene early enough—against the “small clusters” of data points low on the line—instead of letting things fester until you’ve got “big clusters” with lots of incidents, lots of dead, and stretched resources.

“That model tells us how to intervene,” he said. “If you want to prevent large clusters and therefore some horrible event, you could attack the large cluster. But if you can’t do that, go in against the small clusters. Because those small clusters will eventually build to the larger ones.”

This leads to question of when. But again, the model also gives a time scale for intervention, “a time scale in which you’ve got to infiltrate that small cluster, so in effect you own a part of that large cluster, and so [the large cluster] is probably not going to happen.”

We’re not there yet. Johnson is still trying to suss out the trajectory of Brazilian unrest for IARPA, Syrian rebels are still picking away at the Assad regime, and hackers are still trying to infiltrate national infrastructure. And there’s other asymmetrical conflicts that don’t involve people power—think of the exchange of memos concerning Iran’s nuclear ambitions—that would be useful to predict sooner rather than later. Besides, in the garden of small clusters, how do I recognize I’m in the midst of a blossoming conflict?

“What’s al Qaeda’s algorithm vs the U.S. algorithm?” Johnson asked. “I’m not saying that that exists yet, but it will be interesting to know ahead of time how that will play out. And how will you even detect that that’s going to happen?”

“Prediction,” Spagat mused, “is always going to require some sort of leap of faith.”