The fall of empires as an exponential distribution

By Razib Khan | August 4, 2011 12:57 am

I was alerted to Samuel’s Arbesman’s new paper, The Life-Spans of Empires, by the fact that he pointed to his research on his weblog. Interestingly I’m not the only one who was interested, as after I pointed to it on my link round up a few people asked if they could get a copy of the paper (yes, I almost always send papers if I have access). Luckily it’s a nicely elegant piece of work, basically quantifying what we’ve already probably known qualitatively. There isn’t that great of a value-add to quantification as such, but with a mathematical understanding of a topic one can engage in an algebra of mental manipulations so as to construct models with which one can project other facts. Quantitative information is often an excellent way to generate “free information” from theoretical models. The figure above is the primary result of the paper. Basically Arbesman took a data set which was laying around which measured the lengths of various empires (N = 41), and showed that the rise and fall of these political entities tends to follow an exponential distribution: e−λt . This is an incredibly elegant summation of what we know qualitatively: some empires last a long time, but most do not.

Interestingly the mean length of an empire is 220 years. That’s basically what you’d probably expect from intuition, especially if you knew Chinese history. To the left is a density plot I generated with the data provided in the paper. You can see that the mode and mean are a bit different because of the skewness. One of the interesting points about the exponential distribution is that it implies that the the duration of an empire at any given moment can’t tell you the probability that it’s going to collapse in the near future. The distribution is “memoryless.” In other words, the likelihood of doom striking isn’t greater as time passes. This seems somewhat counterintuitive. After all doesn’t the cohesion and elan of the a ruling caste of a given empire wane as the society slowly lose its vital force? Hasn’t the author read Spengler! Arbesman admits that there are more complex equations which can describe the distribution more precisely, but the exponential formula has only one parameter, so it’s quite parsimonious. But even if we have a first approximation we don’t have a total description.

Like evolutionary process as a whole I’m not convinced that the nature of the current data set is sufficient to deny the shifting background parameters which are operative over time. As I’ve noted before, there are two counteracting tendencies over time in human history when it comes to social & political entities:

– A greater rate of cultural change over time

– A greater cohesion and integrative power of social and political systems (more rapid bounce-backs from collapse, and greater civilizational continuity)

One thing I wanted to do is check to see what the correlation between age of the polity and its duration was. My intuition was that older polities will have greater recorded duration. Obviously there’s just more time for them, but some societies, such as Egypt, were very stable for longer periods of time in far antiquity. When I ran the correlate it was pretty weak, -0.23. Below is a chart which shows the scatter plot and the r-squared (correlation squared):

Here’s the original data:

Western Turk (C. Asia)5820.7
Avar (Europe)5802
T’u Chueh Turk (C. Asia)5500.9
Visigoth (Europe)4702.4
White Hun (Indo-Iran)4601
Toba (China)4401.3
Yuen-Yuen (C. Asia)4000.3
Byzantine (Europe)3953.5
Hun (Europe)3800.8
Gupta (India)3700.9
Liu-Sung (China)3302.1
Ptolemaic (Africa)3232.9
Bactria (Indo-Iran)2000.6
Kushan (Indo-Iran)752
Rome (Europe)04
Saka (Indo-Iran)-501.2
Parthia (Iran)-607
Ch’in (China)-902.9
Andhra (India)-1703.7
Hsiung Nu Hun (C. Asia)-1901
Maghada-Maurya (India)-3000.9
Achaemenid (Iran)-5403.2
Lydia (Anatolia)-6100.6
New Babylon (Mesopotamia)-6100.7
New Assyrian (Mesopotamia)-7000.8
Late Period (Egypt)-7151.9
Phrygia (Anatolia)-7600.6
Urartu (Mesopotamia)-8100.9
Babylon (Mesopotamia)-10002.5
Middle Assyrian (Mesopotamia)-10900.5
Hittite (Anatolia)-13201.3
Hsia-Shang (China)-13504
New Empire (Egypt)-15005
Mitanni (Mesopotamia)-15001.4
Elam (Mesopotamia)-160010
Hykso (Syria)-16500.8
Babylon—Hammurabi (Mesopotamia)-17002
Old Assyria (Mesopotamia)-18001
Middle Empire (Egypt)-20003
Akadia (Mesopotamia)-23101
Old Empire (Egypt)-28005
CATEGORIZED UNDER: Anthroplogy, Geography, History
MORE ABOUT: Cliodynamics

Comments (12)

  1. bob sykes

    I like to quibble about Rome.

    Rome persisted 2200 years as a continuous political entity. Even at 0, it had lasted almost 800 years, although 0 is a pretty good estimate for its maturity.

    The Roman government did not disappear when the western provinces were overrun in the 5th Century. It had already been moved to Constantinople for over a century. And Constantinople was in fact near the geographic center of the Empire and a strategic shipping choke point. The various barbarian tribes that conquered did so after they failed to conquer the east.

    I don’t know anything about the history of the other empires, but I wouldn’t be surprised if there are other debatable time lines in the set.

    Alarmists on the right like to compare the US to Rome and infer imminent collapse. We should be so lucky. If we are following the Roman example, we have another 2000 years of superpower dominance. Even if we are where Rome was in its corrupt maturity at 0, we’d still have another 1500 years to go.

    By the way, the scatter plot is crap. The alleged correlation depends on one point.

  2. John Emerson

    I’m very skeptical. The term “empire” covers entities of very different types. Steppe empires are a different thing than sedentary empires. Some empires are large and some small. The Visigothic empire was no more than a large nation (Spain and part of France), and it lost most of its French possessions early and was on the defensive (against Franks, Berbers, and Greeks) for most of its existence. Are Rome and Byzantium one or two? Are the Toba and the Liu-Sung empires or just Chinese dynasties with (originally) foreign rulers? These two were not extensive; both ruled only part of China. There are also differences between the degree of internal integration of various empires, which is a function in part of the technologies of transportation and communication, which changed over time.

    The Khwarizmian empire conquered by Genghis Khan (not listed) had an enormous scope: most the Turkish Republics of the USSR, Afghanistan, much of Iran all the way to the Persian Gulf, and even a foothold on the Arabian peninsula, but almost none of that territory was under firm control (which was why that empire disintegrated and collapsed within two or three years of the first attack.)

    I have these same doubts about the theories of Toynbee and Andre Gunder Frank, and probably Wallerstein (who I haven’t read much of).

    I don’t see the point of mathematizing such mushy data. Leaving it at just “some empires last a long time, but most do not” strikes me as enough. x

  3. AG

    Interesting, life expectancy of democracy historically was about 200 years also. We are at scary moment!

    Take easy, had live under dictationship before, it is not that bad. Actually it help you focus on the things you have better control in life.

  4. mapper

    Interestingly the mean length of an empire is 220 years.

    This point has been made repeatedly by many people. Am0ng historians there was Gumilev and then there was Glubb Pasha. The Fate of Empires by Sir John Glubb:
    (Highly recommended! Brief and very much to the point).

    With the boundaries of any empire being very much subjective matter, with little tweaking almost any function can be fit. So the neat exponential decay only means the author badly wanted it. What really matters is simply what you said: most empires don’t last very long (around 10 generations typically). By the way, you probably don’t want lambda on both sides of the exponential decay equation.

  5. Looks like a power law distribution, which is what you would expect.

    Quetion: To estimate how much time we have in our own civilization, would the principle of insufficient reason be a valid way to think about it? The basic idea, if I remember, is that since we don’t have any idea of when it might end we can say with 95 percent confidence that we are not at the very end (last 5%) of our “natural” lifespan but somewhere in the middle. I believe the size and shape of that interval can be calculated even though it tells you little or nothing.

  6. Peter Ellis

    I’m impressed by your second graph: there appear to be a significant density of empires with duration less than zero. Returning to the first graph – how short does the duration of an empire have to be before they’re frequent enough for us to have one each? Does this perhaps tie into Warhol’s conjecture that we shall all be famous for 15 minutes?

  7. Zora

    David Christian noted that empires seem to get bigger as they approach recent times, and explains this as the slow accumulation of techniques and infrastructure for communication and bureaucracy. Ruling elites, considered as parasites, became more efficient parasites.

    He may not have been the first to observe this; I’d have to rummage in the footnotes to his big book to be sure.

  8. iron0037

    1) The Ptolemaic empire should start at -323, not 323.
    2) I’m pretty sure the Byzantine Empire duration is a typo and should 13.5 centuries instead of 3.5. No significant dynastic changes take place around 750AD.
    3) Where is the Frankish/Holy Roman Empire, the British, the Spanish, the Mongols/Golden Horde, etc? Lots of empires appear to be missing.

    In general, I think it’s tricky to distinguish between an “empire” and a dynasty. Are the different Chinese dynasties truly different empires, or are they a continuation of the same empire under new management? To Bob’s comment, you could argue that the Roman and Byzantine empires are the same thing.

  9. Definitions are pretty important here. Empire length is not a term with a natural meaning. Where do you draw the line between empire and a mere state? Where do you draw the line between successive governmental arrangements in similar geographic regions – between regimes, between dynasties, when there is a major change in territory, when there is an interreginum?

    Similarly, empires typically start from some core that is initially not an empire, expand to the point where they deserve the name, and then do one of several things: contract, split into regional polities, or utterly collapse into widespread anarchy. The first two possibilities are much more common. The last remnant of the Roman Empire, which was the Anatolian rump Byzantine empire, lasted many centuries beyond the collapse of the Western Roman Empire. The Holy Roman Empire of Charlemagne or the Austro-Hungarian Empire, or the Islamic empire at its early peak didn’t just disappear, they fractured into successor states that were sizeable and formidable in their own right. Some of the Mesopotamian and Chinese dynasties arguably pretty much took over more or less intact states from each other (a bit like the successive French regimes that are currently on their Fifth Republic but are in clear succession to the state).

    A more complex, but arguably more sensible way to think about empires is like atom smashing reactions. Some small bits gather lots of momentum, crash into each other making something much bigger that lasts for a while, and then start a decay chain that branches and branches and branches for a long time.

    For example, one fairly productive way to think about state structure and politics in the Middle East and West Asia today is to see the last century and a half or so is as the continued long playing out of the issue of Ottoman Empire succession (arguably even the conflicts in Bosnia and Kosovo can be extended to fit this model).

  10. John Emerson

    The Holy Roman Empire was a bizarre curiosity which existed, more or less, for a thousand years. The Spanish and Austrian Hapsburg empires , nominally part of the HRE, were much more real.

    The significance of the British Empire was far greater than its territorial possessions would suggest, since it controlled sea tarde and to a considerable extent, credit.

    The Portuguese and Dutch empires nominally lasted until WWII.

    I still say GIGO.

  11. With the boundaries of any empire being very much subjective matter, with little tweaking almost any function can be fit. So the neat exponential decay only means the author badly wanted it.

    he seems like a good guy. but you are right that a little conscious or unconscious manipulation can lead to what you say occurs. i just give him the benefit of the doubt (granted, he could of just looked for a set of estimates in a bunch of random books, and found one that fit the exponential).

    I don’t see the point of mathematizing such mushy data.

    your criticisms of the data set are fine. but you could do the same with verbal models. the main difference with mathematical ones is that your critique is hard to avoid or side-step by claiming misunderstanding. i think that’s the value of mathematization. the main downside is excessive confidence in the value of numbers as being precise and without huge margins of error.

    . Even at 0, it had lasted almost 800 years, although 0 is a pretty good estimate for its maturity.

    next you’ll be telling me how many thousands of years a given sumerian king lived? i doubt roman history in the early centuries came out of nothing, but i think it is prudent to chop off the first three years as too mythical to count. even in the period after 500 rome is on the margins of history for centuries.

  12. Rome was very dodgy for the kings and most of the republic. Carthage was only the last of the many enemies that damn near wiped it out. It’s only in the last century BCE under Pompey and Julius Caesar that it started to look like an empire. Zero seems like a reasonable date for the start of the Roman Empire.


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About Razib Khan

I have degrees in biology and biochemistry, a passion for genetics, history, and philosophy, and shrimp is my favorite food. In relation to nationality I'm a American Northwesterner, in politics I'm a reactionary, and as for religion I have none (I'm an atheist). If you want to know more, see the links at


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