Unbelievable numbers of Roman emperors didn't die from natural causes. This is not news, it's actually ancient history.
Scientists have discovered a new mathematical pattern in these untimely, often violent deaths: A power law that describes the fates of those who were left with an entire empire.

"Though they may appear random, power-law distributions of probability are found in many other phenomena related to complex systems," Francisco Rodrigues, a data scientist at the University of So Paulo, Brazil, says. He also notes that the reigns of Caesars are an example of such a context.

Rodrigues claims that the power law distribution which generally determines the longevity of Roman Emperors is the Pareto principle.

The Pareto rule is also known as the 80/20 Rule. It is often concerned with economic inputs, outcomes and outcomes. However, in terms of probability distribution it can be simplified to say that common occurrences have approximately 80 percent probability while rare events have around 20 percent.

This is the case with Roman emperors. Violent ends are more common than natural causes, especially in the early days and years of the Western Roman Empire.

Researchers found that rulers had only a 1 in 4 chance of living to see their first emperor Augustus, who died in 14 CE, through to Theodosius, who died in 395CE).

The Roman Empire's entire arc can be taken into consideration, but it does not include the period from Augustus to the end of the Byzantine Empire (aka, the Eastern Roman Empire) which lasted until 1453 CE.

When you consider the reigns of all 175 Roman Emperors over a longer period of time, each ruler had only a 30% chance of living to an old age (and was not assassinated before).

Researchers found that some years were more dangerous than others during these perilous reigns.

Rodrigues states that when we looked at the time to death of each emperor, the risk was high when an emperor assumed the throne.

"This could be due to the difficulty and demands of the job, and the lack of political competence of the new emperor."

The chances of survival in the top job were greatly improved if emperors made it to the end of their probation period without being beaten by their coworkers. Researchers found that they did tend to do so, at least up to a point.

After the 13-year reign of the Emperors, the risk of their untimely deaths shot up again. This may have been due to the inability of their ambitious and sometimes murderous allies, if not their direct enemies, to show patience.

Rodrigues explains that "it may be that the emperor's competitors concluded they were unlikely to ascend, by natural means, after the 13-year period."

"Perhaps his enemies have regrouped or new rivals might have emerged."

Mathematically speaking, the lives of Roman emperors are somewhat similar to earthquakes. The researchers suggested this in their paper by comparing the likelihood for short imperial reigns (as compared to long ones) with the likelihood of small earthquakes (which tend to be more frequent than big earthquakes).

Both emperors as well as earthquakes seem to cause massive, landscape-altering reverberations. This is not the main point.

These findings were published in Royal Society Open Science.