In this section, we investigate the temporal signature of this seemingly haphazardous stochastic process that is the violent death of a Roman emperor. We also examine whether there is some structure underlying the randomness of this process or not, and we discuss parallels with results in reliability engineering.

Non-parametric and parametric analyses, and interpretation

The data in Table A1 is treated with the Kaplan–Meier estimator, and the results are provided in Fig. 2. The longest rule was that of Augustus, who established the principate, as the early empire was called. He ruled for 45 years, and since he died peacefully of old age, he contributes censored data to the analysis. The results in Fig. 2 are shown up to this last time-to-violent death.

Fig. 2 Survivor (reliability) function of Roman emperors as a function of their time in office. Full size image

Figure 2 reads as follows: at the 3-year mark for example, an emperor had 64% chance of not having met a violent death; at the 7-year mark, those chances drop to 50%, a mere coin toss. The likelihood of a violent death is the complement of these figures, 36% and 50%, respectively.

What does this mean? Three salient features in this figure are important to note:

i. Emperors faced a significantly high risk of violent death in the first year of their rule. This risk remained high but progressively dropped over the next 7 years. This is reminiscent of infant mortality in reliability engineering, a phase during which weak components fail early on after they have been put into service, often because of design or manufacturing defects for example. Roman emperors therefore experienced a form of infant mortality; ii. The reliability or survivor function stabilizes by the 8th year of rule. The emperors could lower their guard a bit if they made it to 8 years… iii. … but not for long: the risk of violent death picks up again after 12 years of rule. This suggests that new mechanisms or processes drove another round of murder. This is reminiscent of wear-out period in reliability engineering, a phase during which the failure rate of components, especially mechanical items, increases because of fatigue, corrosion, or wear-out. Roman emperors therefore also experienced wear-out mortality.

A Weibull plot for the previous nonparametric results is provided in Fig. 3. The plot displays ln{−ln[S(t)]} as a function of ln(t). If the data points obtained are aligned, it can be concluded that the data effectively arises from a Weibull distribution, that is, the underlying parametric distribution giving rise to this violent-death process is indeed a Weibull. The Weibull survivor function is widely used in reliability engineering and survival analysis because it is a highly flexible parametric model, and it is given by

$$S\left( t \right) = {\mathrm {exp}}\left[ { - \left( {\frac{t}{\theta }} \right)^\beta } \right]$$ (3)

β is termed the shape parameter, and θ the (temporal) scale parameter or characteristic life. A shape parameter β < 1 is indicative of or reflects the prevalence of infant mortality in the items under study, whereas β > 1 indicates the prevalence of wear-out failures (decreasing versus increasing failure rates, respectively). Equation (3) is equivalent to a linear Weibull plot (Eq. (4)):

$${S}\left( {t} \right) = {{\mathrm {exp}}}\left[ { - \left( {\frac{{t}}{\theta }} \right)^\beta } \right] \Leftrightarrow \ln \left\{ { - \ln \left[ {{S}\left( {t} \right)} \right]} \right\} = \beta \ln \left( {t} \right) - \beta \ln \left( \theta \right)$$ (4)

Fig. 3 Weibull plot of the survivor function of Roman emperors, and linear least square fit (R2 = 0.962). Full size image

It is interesting that a stochastic process as unconventional and haphazardous as the violent death of a Roman emperor—over a long four-century period and across a vastly changed world—has a systematic underlying structure, and is remarkably well captured by a Weibull distribution. The fact that this result is completely otiose does not diminish its uncanniness!

Why this underlying structure? In statistical theory, the Weibull function is an extreme value distribution, which captures the minimum value of a large collection of random observations. To clarify this point, consider for example a system with a large number n of components placed in series to fulfill a specific function. The failure of any one component results in the failure of the system, its function is no longer provided. The time to failure of the system is therefore the minimum time to failure of any one of its component. Statistical extreme value theory tells us that regardless of the underlying failure distribution of the components, when n is very large, the time to failure of the system approaches a Weibull distribution. Notice in this example the difference between component level and system level considerations, and how the result at the aggregate system level is independent of the failure distribution of any one component. Extreme value theory is also applicable in another related context: consider a single monolithic item. There are no components in this item. But assume that there are n different competing failure processes of this item, whichever one occurs first breaks the item. When n is very large, this will also result in a Weibull distribution of the time to failure of the item regardless of the distribution of each failure process.

The extension of these observations to the violent death of Roman emperors has to be done cautiously. But they offer nonetheless a fruitful venue for exploration. The fact that the time signature of the stochastic process of interest here is remarkably well captured by a Weibull distribution suggests that it is perhaps indeed the result of a very large number of underlying processes conspiring to violently eliminate the emperor. The fact that there were many pathways to the violent death of an emperor, with large numbers of individuals and motivations for undertaking the grisly task, makes the Weibull, an extreme value distribution, theoretically plausible in this case.

Mixture Weibull distributions

A closer inspection of Fig. 3 shows two distinctive slopes for the data points, before and after ln(t) ≈ 2.5, which corresponds to the onset of the wear-out failures seen in Fig. 2. A mixture Weibull distribution is therefore fitted to the data, and the maximum-likelihood estimates of its parameters are provided as follows:

$$\begin{array}{l}\widehat S\left( t \right) = 0.876 \cdot {\mathrm {exp}}\left[ { - \left( {\frac{t}{{12.835}}} \right)^{0.618}} \right] + 0.124 \cdot {\mathrm {exp}} \left[ { - \left( {\frac{t}{{14.833}}} \right)^{13.387}} \right]\end{array}$$ (5)

Equation (5) provides an analytical confirmation of the previous observations, that Roman emperors experienced both infant mortality (β = 0.618) and wear-out mortality (β = 13.387) in the form of violent death. This parametric result is shown in Fig. 4.

Fig. 4 Mixture Weibull survivor (reliability) function of Roman emperors, and the nonparametric results. Full size image

The emperors who experienced infant mortality were not unlike engineering components that suffer early failures after they are put to use: weak by design or fundamentally incapable of meeting the demands of their environment and circumstances. Examples from each century abound, for example Galba (d. 69 CE), Pertinax (d. 193 CE), Macrinus (d. 218 CE), and Severus II (d. 307 CE). These were times of upheaval, and in the first two cases, these turned out to be times of transition to new dynasties (the Flavian, and the Severan, respectively). Emperors’ infant mortality can be seen, in part, as both causes and consequences of times of crisis and instabilityFootnote 7.

The emperors who experienced wear-out mortality met their end through different failure mechanisms. Consider first that some engineering components experience an uptake in failures (wear-out failures) after they have been in service for a long time. They may have been sturdy at first and benefitted from clement operational environments to start with. But through degradation, fatigue, or increased harshness in their operational environment, they begin to experience wear-out failures. The emperors who survived the first 8 years of their rule, as seen in Fig. 2, had a grace period of about 4 years. Violent death came to them afterward (wear-out mortality) because, for instance, their old enemies had regrouped or new ones emerged, because they had alienated an increasing number of parties, or because new weaknesses in the imperial rule appeared or grew. These new murderous processes clearly had a different temporal signature than those driving the emperors’ infant mortality, as seen in Fig. 3 and in the different characteristic life parameters of each Weibull distribution in Eq. (5). For example, the death of Domitian after a 15-year rule (d. 96 CE), or Commodus after a 12-year rule (d. 192 CE), or Gallienus after a 15-year rule (d. 268 CE) are illustrative of wear-out mortalityFootnote 8.

The failure rate (Eq. (2)) of the parametric fit (Eq. (4)) is given in Fig. 5. The result shows a remarkable bathtub-like curve, a model widely used, and empirically confirmed in reliability engineering for a host of mechanical and electronic components. Roman emperors, like these engineering items, therefore experienced a bathtub-like failure rate.

Fig. 5 Failure rate of Roman emperors (parametric fit of the time-to-violent-death). Full size image

The results in Fig. 5 lends themselves to an interesting interpretation:

i. The decreasing failure rate early on, the signature of infant mortality, reflects as noted previously a prevalence of weak emperors who were incapable at the onset of their rule to the handle the demands of their environment and circumstances. The fact that the failure rate was decreasing though suggests a competition between antagonistic processes, on the one hand those that sought to violently eliminate emperors (elimination), and on the other hand those that reflected the emperors learning curve to better protect themselves and perhaps eliminate their opponents (preservation). Examples abound in Roman history of this competition. Up to the first 12 years of one’s rule, the preservation processes steadily improved their performance, and the situation can be casually summarized as “whatever didn’t kill them [the Roman emperors] made them stronger” or less likely to meet a violent death; ii. The increasing failure rate after 12 years of rule, the signature of wear-out failures, reflects as noted previously an uptake in failures through degradation with time, fatigue, or increased harshness in their circumstances. A growing mismatch between capabilities and demands under changing (geo-)political circumstances. This can be due to a number of reasons discussed previously. The fact that the failure rate was increasing after this 12-year mark suggests again a competition between the same antagonistic processes noted in (i), and this time the preservation ones were on the losing end of this competition. This result can be causally summarized as “whatever didn’t kill them made them weaker” after a 12-year rule.

Beyond these specific details, what does it mean to find a coherent structure within a stochastic process of historical nature as the one here examined? Roughly speaking, the result implies the existence of systemic factors and some level of determinism, in an average sense or expected value, superimposed on the underlying randomness of the phenomenon here examined. In other words, the process is not completely aleatory; it has some deterministic factors overlaid on its randomness. Conan Doyle, in Sherlock Holmes: The Sign of Four, expressed this general idea rather accurately when he wrote:

While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to.

The results in this section suggest a similar idea underlies the violent death of Roman emperors.

Etiology and suggestion for future work

The previous subsections investigated the temporal signature of the phenomenon here examined, the violent death of emperors, a spectacle of brutality and violence not unlike the gladiatorial games, except it stretched over four centuries and affected the entire Roman world (Millar, 1977).

What has not been explored is the etiology or causal basis of this phenomenon, or why emperors repeatedly met violent deaths in the first place, not just temporally how. The immediate causes of violent deaths of Roman emperors are frequently discussed in the literature. They can be found for example in Scarre’s (1995) “Chronicle of Roman Emperors”, and a short summary is provided in Retief and Cilliers’ (2005) “Causes of death among the Caesars (27 BC–AD 476)”. The entries include statements such as “murdered by the sword/dagger […]”, “poisoned by [name of individual]”, or “decapitated by the soldiers”. These explanations are of little interest, and they do not reflect the complex nature of causality in this context. The causal basis of the phenomenon here examined intersects a number of fundamental issues in Roman history, the development and pathologies of the Roman monarchy for example, the problem of imperial succession, the role of the praetorian guard, the loyalties of the legions, and the geographic extent and resources of the empire, to mention a few. These issues and the complex nature of their relations with the phenomenon here examined are left as a fruitful venue for Roman historians to examine. It is worth noting that the the spectacle of regicide of Roman emperors is related a reciprocal way, as both a causal factor and a consequence, to the decline and fall of the Roman empire. As such, it deserves careful attention in future work.