This is Jerry McManus' follow-up to his very insightful article, Modeling Collective Behavior, and it delves into the scientific meaning of a commonly-used, but sometimes misunderstood term of complex systems theory - "tipping point". I have briefly added to his discussion of tipping points in forest ecosystems and financial systems by referencing a recent report produced by Goldman Sachs on the Eurozone crisis (marked within the text). Other than that, it's all Jerry! So, without further ado...

Modeling Tipping Points

Intro

The term "tipping point" crops up frequently, especially in discussions of world events such as financial collapse and climate change. Like the word "sustainable" which gets used with increasing abandon, and often with little thought to the actual meaning of the word, so too the term tipping point can be elusive and used in contexts where it may or may not be suitable. I've probably been guilty of this myself, but recently my thinking on the subject has been greatly clarified and I hope to pass on what I've learned about what does, and does not, constitute a tipping point.

We will look at a couple of simple models that explore tipping points in an attempt to think more clearly about the problems of financial contagion and climate change, but first we must try and nail down a working definition of tipping points.

Straw, meet proverbial camel

So, what exactly is a tipping point? A tipping point can most easily be defined as a small change that can have a large effect on the end state of a system. For example, we have the well known proverb of the straw that breaks the camels back. In this scenario a linear increase in the weight of straw loaded onto the back of a camel leads beyond a certain point to an abrupt change in the health of said camel.

 
Fig 1. Straw weight plotted vs number of straws is a linear function. Health of camel vs straw weight crosses a direct tipping point in a step function.

That is a classic example of a small change, in this case the addition of one straw, crossing a direct tipping point and causing a disproportionately large effect by suddenly and completely disabling our unfortunate camel.

Next, we see another familiar example of what looks like an abrupt change, world population growth:

 
Fig 2. World population charted on a long time scale. Is this an example of a tipping point?

This curve, shaped like a hockey stick, is very familiar to those who study exponential growth and it would be easy to conclude that where we see a distinct kink in the curve there must have been some sort of a tipping point beyond which the population exploded. But is it really a tipping point? Not necessarily. One feature of exponential growth curves is the fact that the shape of the curve can change dramatically depending on your time frame. Zoom in far enough on the scale and what looked like a sharp kink becomes much more gentle, almost linear.

 
Fig 3. The same exponential functions charted at different scales. Only at the larger scales does the kink in the curve become pronounced enough to be mistaken for a tipping point.

Recall our definition of a tipping point: a small change that can have a large effect on the end state of a system. Kinks in curves on the other hand are not necessarily tipping points, they are just the result of exponential growth as seen from a sufficiently large perspective. Unlike the proverbial straw and our pitiful camel, there really is no point in the history of world population where we can say the addition of one more person suddenly caused the population to explode.

The domino effect

It's easy to see that our world has grown increasingly complex, especially since the industrial revolution and the introduction of massive energy subsides in the form of fossil fuels. Not just the explosion in population, but also the dramatic growth in almost all aspects of human society. The amount of resources we use, the number of artifacts available to help us consume those resources, the amount of waste produced by those artifacts, the size and complexity of our political and financial systems..., the list goes on.

People hold a variety of viewpoints on the future of these increasingly complex systems, from the extreme optimism of unbounded technological progress to the extreme pessimism of imminent catastrophic collapse followed by a future of post-apocalyptic hardship. However, as the inherent un-sustainability of our current system slowly becomes ever more apparent, seldom more starkly so than the near total collapse of the financial system in 2008, then so too the question of risk becomes vitally important.

Risk can be hard to quantify, especially catastrophic risk, which makes it all the more interesting that one way to approach that question is to model density. For example, picture a chessboard. Now randomly place some dominoes in an upright position on that chessboard, not too many, just a few scattered around. Knock over any one domino and there is little risk that it will knock over many of the other dominoes, maybe one or two, but certainly not all of them.

 
Fig 4. Low density of dominoes means less risk of contagion

Now place a large number of dominoes on the board. Knocking over any one domino will certainly knock almost all of them over. The risk has increased dramatically with the number of dominoes. Is this a simple linear relationship? Or is there a tipping point beyond which the addition of one more domino spells almost certain disaster?

 
Fig 5. Increase the density and the risk also increases, but is it a linear relationship?

Tipping points and firestorms

To help us explore this question we will turn to the percolation model familiar in physics. This model has good "fertility" which simply means that while it originated in the study of fluids percolating through a substrate, such as water through soil, it can also be applied with remarkably good results to other fields, such as forest fires.

For this example we turn once again to agent based modeling, for an introduction to this topic see the post Introduction to Agent Based Models. First, as in the other models, we start with a simple grid. We then randomly populate that grid with a forest at a given density expressed as the percentage of grid cells occupied by a tree, one tree per grid cell. Once again we apply one very simple rule based behavior:

1) Fire can only spread between immediately adjacent trees.

Now we can construct a simulation where a fire starts along the entire length of one edge of the grid and then we ask the question: How likely is it that fire will spread, or "percolate", from one side to the other?

Setting the density of trees to different values reveals a surprising result. The risk of catastrophic fire does not increase in a linear relationship with the density of the forest. Instead there is a tipping point at about 59% density.

 
Fig 6. A distinct tipping point in the percolation model at about 59% density. Beyond this threshold the risk of a firestom rises almost vertically to near 100 percent.

Below this value the chances are very low that the fire will spread very far as there simply is not enough paths between adjacent trees for the fire to follow. Set the density higher than that threshold, by even a small amount, and the risk of the fire spreading across the entire forest rises dramatically.

 
Fig 7. On the left, a random arrangement of trees at a density of 57% results in fires that consume only about 15% of the forest at most. On the right, increase the density by just a few percent to just above the threshold of 59% and it suddenly becomes much more likely the fire will spread to the other side and consume two thirds of the forest or more.

Tipping points and financial contagion

This model can also be applied with good results to other collections of entities connected by network paths, such as our increasingly complex financial system.

 
Fig 8. A typical network diagram. Like the forest, increase the density of connections and you risk crossing a threshold beyond which you greatly increase the chance that the entire system can collapse.

From this perspective, applying what we learned from the percolation model, it's easy to see that the higher the density of network connections then the more likely it is that a threshold will be crossed that greatly increases the risk of a collapse. The quants and other financial gurus behind the scenes at the mega-banks may have thought that creating the great, billowing, quadrillion dollar bubble of derivatives and other exotic couterparty casino bets would reduce risk by spreading it around the globe. Instead, it's much more likely that just exactly the opposite is true. Beyond a certain tipping point in the density of couterparty risk the odds of a catastrophic financial meltdown rises dramatically, just as we saw in 2008.

Ashvin: In one of its reports, Goldman recently analogized the Euro area to a dense forest, in which the risk of "financial fires" spreading extensively is very high and has only been made higher by policy responses. Perhaps one of the reasons EU leaders continue to create sovereign guarantees and contingent liabilities (debts) to address a sovereign debt crisis in a closed region is because they fail to understand the nature of tipping points that Jerry has clearly outlined above. Here is an excerpt from the report, courtesy of ZeroHedge:

Goldman on Europe: "Risk of 'Financial Fires' is Spreading"

 

4. Too Much Combustible Material Still Around

 

The Euro area is a financially closed region, with more than 85% of sovereign bonds held by residents of the area. If we add to this the fact that most claims against governments are held by financial institutions domiciled in the area, the risk of 'financial fires' spreading is high. The prevailing policy view that bigger 'firewalls' would make investors more comfortable about purchasing sovereign bonds of EMU countries. This is predicated on the idea that the existence of a funding backstop would prevent credit shocks in one of the EMU members from spreading to other issuers. That said, we doubt the current infrastructure can produce the same effects on markets as the ECB's long-term liquidity injections (LTROs). Our view is based on the following considerations.

 

Size: Even if we combine the full uncommitted capacity of the EFSF and the ESM (EUR700bn), the total would not be sufficient to backstop the bigger markets of Spain and Italy. The former's borrowing requirement (amortization plus deficit) over the next two years is EUR305bn, while the latter's amounts to EUR525bn.

 

Seniority: The ESM holds 'preferred creditor status' over existing bondholders (art.13 of the Treaty establishing the ESM). In practice, this means that if the facility is used to provide an EMU member country under conditionality, it would subordinate existing bondholders (twice, if the IMF also participates in a bailout). Given that investors are aware of this, they would require compensation to bear such risk. This could exacerbate, rather than mitigate, a crisis.

 

Governance: The existing vehicles cannot intervene pre-emptively in markets at signs of tension. Rather, they would be activated only after a full crisis has erupted. The procedure envisages that the ECB would ring an alarm bell should tensions threaten the stability of the Euro area. The sovereigns experiencing tensions would need to formally ask for help, and sign a memorandum of understanding, before any financial support can provided. Admittedly, a 'fast track' option is also available, based on 'light conditionality' and allowing the EFSF to intervene in secondary markets. Still, the fixed size of resources could raise questions on the effectiveness of the operations.

The truth is that the entire world, and especially the Euro area, is well past the point at which credit contagion can be isolated to a few minor locations and be extinguished with ease. Jerry has skillfully demonstrated the scientific foundation of this truth underlying not only our global financial system, but perhaps many of our environmental systems as well. And now I will send it back to Jerry for discussion of the latter.

Tipping points and climate change

Jerry: Few topics have been more liberally peppered with talk of tipping points than that of climate change. If you ask those people not in denial about global warming what the threshold for climate catastrophe is I suspect many will answer without hesitation "greenhouse gases" (GHG's), or perhaps "CO2". This is understandable, after all while climate science is certainly a complex topic it has been established that GHG's are the main drivers of global warming due to the well known and unambiguous physics of Earth's atmosphere. Thus, at least when it comes to public policy, it is relatively easy to get ones head around a key number like CO2 concentrations in parts per million.

Unfortunately, I believe this is a dangerous fallacy.

First, recall once again our definition of a tipping point: A small change that can have a large effect on the end state of a system. Due to our prodigious burning of hydrocarbons in the form of fossil fuels the amount of CO2 in the atmosphere is indeed growing exponentially, some would say super-exponentially, which is to say the rate of change is itself accelerating. But remember, a kink in a curve is not necessarily a tipping point. There really is no point at which we can say that adding one more part per million of CO2 will trigger climate catastrophe but below that level and, oh thank god, we are safe.

This is especially true when you consider the residence time of CO2 in the atmosphere which is measured in decades, if not centuries, due to the fact that the portion not absorbed by oceans or sequestered by buried vegetation is removed only by extremely slow geologic processes. The simple fact is, even if starting tomorrow all seven-billion-going-on-nine-billion people on Earth stopped emitting any CO2 whatsoever we would still suffer a de-stabilized climate and rising sea levels for decades to come due to the carbon we've already spewed into the atmosphere. This dynamic is greatly exacerbated by the extremely large thermal inertia of the world's oceans and icecaps. Put another way, we are probably only just now feeling the full effects of carbon that was emitted decades ago, with much worse to come. Such is the pernicious effects of very long feedback delays in a complex system.

The climate is a dynamic system that can be said to exist in an unstable equilibrium. This equilibrium is variable, the complex interactions of sun, sea, and atmosphere give rise to oscillations that are measured in decades and, in the case of ice ages and inter-glacials, millennia. Within that range however there has formed a kind of homeostasis that is conducive to life, a balance between the positive feedbacks that create reinforcing loops and the negative feedbacks which create regulating loops.

It is in this balance where the greatest danger of a tipping point can be found. Should any of the factors that contribute to positive feedbacks get too far out of bounds then this can tip the balance in favor of self-reinforcing behavior that, once started, can quickly take on a life of its own and overwhelm our capability to regulate it which will lead inexorably to accelerating or "runaway" climate change.

Melting permafrost. Dying forests. Acidic and anoxic oceans. Vanishing ice-cap albedo. These are just a few of the factors at the heart of global climate feedback loops, any one of which could be dangerous, but taken all together and greatly magnified by feedback delays measured in decades if not centuries then it becomes clear that the crisis we face completely dwarfs the problem of CO2 concentrations alone.

We are altering both the chemistry of the atmosphere and the composition of the biosphere at a rate orders of magnitude greater than that seen in the geologic past. At this point even cutting CO2 emissions to zero would be woefully inadequate, we would still need to take desperate measures in an attempt to restore the previous balance by putting in place global negative feedbacks. Reforestation, carbon sequestration, cloud seeding, all this and more while at the same time we power down and depopulate to levels last seen many decades ago. Unfortunately, given both the enormous challenge of such an undertaking compounded by the very long feedback delays in the climate system we would probably need to have started such a program many years ago. We may have already passed the tipping point of no return.

Conclusion

I hope this small excursion clears up some of the confusion surrounding the subject of tipping points. It's important first of all to understand what a tipping point actually is. Like others, I myself have often confused exponential growth as a tipping point when in truth it is really only a kink in a curve as seen from a sufficiently large perspective.

Next, it's important to understand the difference between direct and contextual tipping points. The example of the straw that broke the camels back is a direct tipping point, whereas the example of the forest fires is a contextual tipping point. Increasing the density of the forest by one more percent did not directly cause a firestorm, but it did cross a critical threshold where the chances of a firestorm greatly increased. Beyond that point all that was needed was a spark for catastrophe to follow, much like the subprime housing crisis was the spark that melted down the global financial system.

Finally, let us not become beguiled by talk of CO2 concentrations in the climate crisis. The real and present danger to countless future generations, if not all life on Earth, are positive feedbacks that are poised to cross a critical threshold (if they have not done so already), at which point they take on a life of their own and race away from our capability to damp them down and return the climate to the happy equilibrium we've enjoyed for these many millennia.

Further reading

A good presentation about climate change and positive feedbacks
Beyond the Tipping Point