I have recently been reading up on cliodynamics, which is the application of mathematical models to historical population sociodynamics. If this sounds familiar, that's because it's exactly what I've been doing for the last few months. So it's good to put a name on it.
One thing that particularly interests me is the theory of social collapse. This is something that's been missing from my own work, try as I might. I think that cliodynamics might be able to help me answer this question.
In one of his models, Peter Turchin defines a measure called
asabiya, which is an Arabic term implying social cohesion. Turchin applies this in a simulation by logistically raising the asabiya of a hinterland/border cell to 1, while internal cells exponentially decay to 0. If the average asabiya of a whole hegemony reaches a critical value, the entire thing collapses. \[S_{t+1} = S_t + r_0 S_ t (1  S_t)\] \[S_{t+1} = S_t  \delta S_t\]
The effect of this is that empires should only last about 200 years on median, which is much closer to the real world than the massive empires I've been dealing with. Once a critical mass is reached, my empires last...forever. There's no real pressure to collapse, barring a random extinction event such as a plague.
Another
paper in the same vein explores secession and hierarchy elements in a manner that is very similar to mine. So I'm happy to synthesize it into my body of knowledge.
However, I still need to account for the size to asabiya feedback mechanism. In all these models, the size of the polity $A(t)$ is part of a larger model which is affected by $S(t)$ (the asabiya), or even the metaasabiya $R(t)$. My own $A(t)$ is determined by the growth of cities and their infrastructure reach.
Therefore, I'm going to pull in
another model developed by Turchin to correlate population $N$, state resources $S$, and internal warfare $W$. This provides a way to "hook into" the $K$ carrying capacity mechanism and thereby connect the two models. In particular, we have: \[\dot{N} = r_0 N \left(1  {N \over k_\textrm{max}  c W}\right)  d N W\] \[\dot{S} = \rho_0 N \left(1  {N \over k_\textrm{max}  c W}\right)  b N\] \[\dot{W} = a N^2  b W  \alpha S\]
So as population $N$ varies, it will affect total area $A$ and therefore $S$ (I don't know why he uses the same variable for asabiya and state resources).
The question is how to discretize this function. The model as given is a hegemony/politylevel equation, governing the total state. Therefore, I have to scale W to the individual city:\[\dot{N_i} = r_0 N_i \left(1  {N_i \over k_{i,\textrm{max}}  c W {N_i\over\sum_j N_j}}\right)  d N_i W {N_i\over\sum_j N_j}\]
I also don't have to collapse the entire hegemony at once if the critical point $S_{crit}$ is reached. Normally, secession is controlled by a loyalty variable. I'll still maintain this, but I'll also make secession likely to happen if the average asabiya is below the critical point, regardless of loyalty score. This will cause quick collapse if the empire overextends itself but also provides an opportunity for it to "recover" after that collapse.
First: some testing of the basic asabiya concept. I'll run the model as normal, while keeping track of what the asabiya should be. Then examine which hegemonies should have collapsed. The only change I'll make to the normal model is to suppress other collapsing mechanisms such as surrender and secession.
Initially, I thought that I should use the areadependent asabiya model: \[S_{t+1} = S_t + r_0 \left(1  {A \over 2 b}\right) S_ t (1  S_t)\] to account for the size of the polity. This proved to a very strong feedback effect to the point where $S$ decayed to its endstate nearly immediately.
Examining the largest hegemony (the Thav empire) using the nonareadependent model:

S (red), A (blue): over time 
It quickly becomes clear that there is an effect introduced by Turchin's simulation space which accelerates empire demise. Here, the asabiya of this large empire does not drop even close to the $S_\textrm{crit}=0.003$ value which provides Turchin with such historical variation. I believe this to be due to a few factors.
In these early stages of implementation, there is still no feedback from $S\to A$. Therefore, there can be no collapsing mechanism as $S$ enters a positive (and mutually destructive) feedback loop. There are a couple of places I can connect this loop. The first is in terms of
warfare escalation. If the power differential between two hegemonies is above a threshold, it becomes more likely that the stronger will invade. Individual hex power is defined by the following equation: \[P = A \bar S \exp(d/h)\], where $d$ is the distance from the capital and $h$ is a dropoff factor usually equal to 2. We can average this for every hex in the hegemony to determine the $\bar P$ for the hegemony. In the case of Thav above, $\bar P = 4.72036$. This will also be interesting to examine over time to see if it's useful anywhere else.
The largest hegemony during this 500year run is the Winan empire:
The jump in $A$ appears to be directly attributable to the expansion mechanic kicking in around 335. All of a sudden, the few (4) cities that Winan has conquered begin pumping out colony cities. This dramatically increases the reach of the empire, which has a detrimental effect on $S$. From this and the Thav simulation we can see that it takes between 300400 years for a city to grow to expansion size (under the current rules).
Of course, there is no law in the real world that demands that settlements refuse to send out settlers unless they have more than 5000 people. But these rules are justified elsewhere on this blog.
Looking at overall power:
There's a slight bump during the initial expansive "golden age", but this quickly returns to what appears to be a pretty steady number. The interesting thing is that $\bar P$ appears to be steady regardless of the continued movement of $S$ or $A$.
The other place is in the
infrastructure spread calculation. Remember that as cities grow, they spread their influence, creating new borderlands for the empire (and eventually butting heads with other empires). What if this growth (already affected by terrain) were also affected by the power of the city in question? More powerful cities could spread their influence further and vice versa. In addition, when Turchin applies this model to realworld conditions, he imposes an additional constraint that regions must experience a certain amount of cultural conflict to be considered true "frontiers." For example, linguistic and religious borders contributed to this intensity score. That shouldn't be terribly difficult to implement. I'll try these next time.
Through all this, I feel very much as if I'm exploring an actual history. Although it's shaped by my own decisions and preconditions, on net it's stochastic to feel alien. And that's pretty cool.