The longevity of historical civilizations seems to follow an exponential distribution (Arbesman 2011). In this manuscript, I explore what this empirical insight might tell us about the statistical nature of existential threats that a civilization might face.

Contents

[See a preliminary talk on the topic here]

Civilizational collapse is a complex process. Several reasons can contribute to collapse, including increasing complexity of a society, environmental stresses both endogenous and exogenous to the system, and competition with neighboring civilizations. Furthermore, defining a civilization, and where it begins in time and space, is a problem that has many answers depending on the purpose of your definition.

Setting these challenges aside, Samuel Arbesman (2011) used empirical data of Empire-lifetimes over three millennia to argue that the longevity of civilizations follows an exponential distribution. In the draft of his new book, Anders Sandberg extends this to different datasets of civilizations, confirming the same result. He suggests that by virtue of the exponential distribution's memoryless property, we can conclude that civilizations do not age, i.e., the risk of their collapse is the same irrespective of how mature or young a civilization might be.

From Arbesman (2011): Probability density function of empire lifetimes, based on the lifetimes of 41 empires.

Though this is surprising, from a complex system perspective, civilizations merely join a family of other complex systems such as biological species (Van Valen 1973), firms, and institutions whose lifetimes also fit exponential distributions. But, what might explain the exponential distributions for the longevity of these complex systems? Does the fact that the lifetimes follow an exponential distribution tell us something about the nature of terminal threats these systems face? In this manuscript, I explore these questions with civilizations in mind.

In the next section, I discuss some of the simple ideas in literature that might explain the exponential distribution of civilizational longevity, i.e., why they do not age. After that, I dwell deeper into stochastic processes that might result in such exponential distributions of durations between events (i.e., longevity). Realizing that civilizations face not a single threat, but a multitude of them, we discuss how the *summation* of stochastic variables might alter the effective distribution that we observe. Finally, we show some simple examples of aggregate stochastic processes that lead to exponential distributions. With these results, we demonstrate how we might constrain the probability distribution of threats that result in civilizational collapse.

What is ageing?

Common explanations:

- Red Queen Effect: Species are in competition between co-evolving species constantly and therefore despite their evolution their 'survival fitness' remains constant and hence becomes independent of any intrinsic characteristics (therefore making the distribution of their longevity memoryless, i.e., exponential).
- Potential pitfalls: What about isolated civilizations?

- More generally: the defense capability of civilization increases proportionally to the increase in risks of collapse.
- What mechanism can cause this proportional increase?

- The rate of threats against civilization remains constant on average, and the civilization's defense capabilities are not increasing.
- This is difficult to conceive since the technologically advanced civilization must possibly have a better advantage over lower-technology civilization (many examples for this). So a constant defense capability across the board is unlikely.