A staggering amount of Roman emperors did not die of natural causes. That's not breaking news; it's literally ancient history.

But in those untimely and often violent deaths, scientists have now identified a new mathematical pattern: a power law that describes the fate of so many who died with an entire empire at their feet.

"Although they appear to be random, power-law distributions of probabilities are found in many other phenomena associated with complex systems," says data scientist Francisco Rodrigues from the University of São Paulo in Brazil, noting that the reigns of the Caesars themselves are one such context.

According to Rodrigues, the power law distribution that generally defines the longevity of Roman emperors is what's called the Pareto principle.

Also known as the 80/20 rule, the Pareto principle is usually concerned with economic inputs and outcomes, but in terms of probability distribution, it can be simplified to mean that common occurrences have about 80 percent probability, while rare events have about 20 percent.

In this case, with regards to the fates of Roman emperors, violent ends are the more common events, with death from natural causes being significantly rarer – especially in the early days of the Western Roman Empire.

During that period – from the first emperor Augustus (who died in 14 CE) through to Theodosius (who died in 395 CE) – the rulers only had about a one-in-four (24.8 percent) chance of living long enough to die from natural causes, the researchers found in their new study.

Taking in the complete arc of the Roman Empire, however – from the time of Augustus through to the ultimate end of the Byzantine Empire (aka the Eastern Roman Empire, which extended on until 1453 CE), things didn't get much better.

Once the reigns of all 175 Roman emperors over this longer timeframe are considered, each ruler still only had about a 30 percent chance of living to a ripe old age (and not being assassinated or otherwise killed beforehand).

During these perilous reigns, some years were more perilous than others, the researchers found.

"When we analyzed time to death for each emperor, we found that the risk was high when the emperor took the throne," Rodrigues says.

"This could have something to do with the difficulties and demands of the job and the new emperor's lack of political expertise."

Provided emperors safely made it through their probation period without being killed by their co-workers, survival chances in the top job swiftly improved. Or at least, they tended to, up until a certain point, the researchers found.

Once the emperors had reigned for about 13 years, their risk of untimely death shot up again, perhaps reflecting patience running out in the minds of their ambitious, occasionally murderous allies – if not their outright enemies.

"It may be that after the 13-year cycle the emperor's rivals concluded they were unlikely to ascend the throne by natural means," Rodrigues says.

"Perhaps his old enemies regrouped, or new rivals may have come to the fore."

Mathematically speaking, the lifetimes of Roman emperors were somewhat akin to earthquakes, the researchers suggest in their paper – comparing the likelihood of short imperial reigns (as opposed to long ones) to the likelihood of small earthquakes (which are much more common than big earthquakes).

Both of them – emperors and earthquakes – also appear to trigger massive, landscape-altering reverberations of violence too. But that is perhaps beside the point.

The findings are reported in Royal Society Open Science.