Scale invariant behaviour in avalanches, forest fires, and default cascades: lessons for public policy

We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal "fingerprint" of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.  - Bak et al., 1988 (one of the greatest abstracts ever written!)

1987 saw the publication of an extraordinary paper--one which led to a dramatic change in our understanding of the dynamics of certain kinds of dynamic systems. Most importantly  . . . introduced the concept of self-organized criticality, or self-organization to the critical state--which is a condition neither fully stable nor fully unstable, with a characteristic size-distribution of events (or failures). In the kinds of systems that interest geologists, earthquakes and avalanches were quickly recognized as being SOC systems, and SOC was recognized as the most efficient means of transmitting energy through a system.

Avalanches and SOC

An early computational experiment went like this:  imagine a pile of sand, on which single grains of sand are dropped one by one until an avalanche occurs.  An avalanche occurs when the slope at some local point is greater than a defined value.

If your sandpile is two-dimensional (length and height--imagine a cross-section of a real sandpile), you would have to visualize it as a string of numbers, where each value represented the number of grains of sand stacked at that point. In the figure below, we are only looking at half of the pile, from the midpoint to the edge.

In our simple sandpile consisting of four stacks, a grain of sand of thickness dx falls onto the middle stack. If the difference in heights between this stack and its neighbour x1 in the figure above) exceeds some threshold value n, then one grain of sand would drop from the higher stack onto the lower stack. You would then have to check whether the height of the next stack was now more than  n higher than its neighbouring stack. If so, then another grain of sand would drop down one more stack and so on to the end of the pile.

What happens in a two-dimensional sandpile is that eventually the height of the sandpile is such that each stack is exactly n higher than its neighbouring stack. As a new grain of sand is dropped onto the pile, it migrates along all of the stacks and drops off the edge of the pile.

The behaviour of the sandpile is very simple; but what happens when you move to a 3-dimensional model (I'm counting the height of the pile as a dimension--not all authors describing this problem do so!)? You might expect similar behaviour--that the slope of the pile will increase until a single grain of sand causes a rippling cascade through the entire pile. This doesn't happen, for it would imply that the natural behaviour of the system is to evolve towards a point of maximum instability. In the experiment, the behaviour of the sandpile was much more interesting. The pile built up until it reached a form of stability characterized by frequent avalanches of no characteristic size.



Bak et al. (1987) called this condition of minimal stability the "critical state", and pointed out that as it developed independent of modelling assumptions and external parameters, it arose by self organization--the term "self organized criticality" (SOC) was introduced to describe the process. The characteristics of systems displaying SOC are fractal geometry, and flicker noise (also called 1/f noise).

There are many systems in nature--and increasingly in the human environment--which are similar to the avalanche model described above. Real avalanches, and similar mass sliding events (debris flows in the deep sea, for instance) have been recognized as SOC processes; along with earthquakes, volcanic eruptions, and economic events.

Forest fires were quickly recognized to be characterized by SOC--at least in environments without a lot of active management. Curiously, it quickly turned out that the effects of fire management, at least as practiced in the United States, might have had an effect opposite to that which was desired.

Fire suppression in the United States

“Strange to say, that, obvious as the evils of fire are, and beyond all question to any one acquainted with even the elements vegetable physiology, persons have not been found wanting in India, and some even with a show of scientific argument(!), who have written in favor of fires.  It is needless to remark that such papers are mostly founded on the fact that forests do exist in spite of the fires, and make up the rest by erroneous statements in regard to facts.”   B.H. Baden-Powell

As European settlers spread through what became the United States, they were confronted by an unusual world. Wilderness was something that had to be eliminated so that "civilization" could spread. Forests were to be cut and the land put to the plow. This was more than an economic imperative--it was a moral imperative as well. 


The rapid westward expansion in the 19th century brought railroads, and railroads brought further development and fire. While clearance of the forest was necessary for development, the desire to create a forestry industry based on sustainable harvesting rather than a short-sighted liquidation of old forests was driven by European examples. And thus the American ideas of forestry were transformed by the turn of the 20th century. Forests were resources that had to be tended. And as resources, any fires within them resulted in economic losses.

Fire had been used as a method of maintaining the forest by the native populations--but such a method was far too messy and unpredictable for a modern people--particularly those who looked to the forestry programs of western Europe, where fires were uncommon. The European model worked tolerably well in the eastern forests in North America, where water was plentiful year-round; but this model turned out to be unsuitable for the western forests, the life cycles of which required fire as a controlling element.

Major Powell launched into a long dissertation to show that the claim of the favorable influence of forest cover on water flow or climate was untenable, that the best thing to do for the Rocky Mountains was to burn them down, and he related with great gusto how he himself had started a fire that swept over a thousand square miles. - Bernard Fernow

The forests of the southwestern United States were subjected to a lengthy dry season, quite unlike the forests of the northeast. The northeastern forests were humid enough that decomposition of dead material would replenish the soils; but in the southwest, the climate was too dry in the summer and too cool in the winter for decomposition to be effective. Fire was needed to ensure healthy forests. Apart from replenishing the soils, fire was needed to reduce flammable litter, and the heat or smoke was required to germinate seeds.

In the late 19th century, light burning--setting small surface fires episodically to clear underbrush and keep the forests open--was a common practice in the western United States. So long as the fires remained small they tended to burn out undergrowth while leaving the older growth of the forests unscathed. The settlers who followed this practice recognized its native heritage; just as its opponents called it "Paiute forestry" as an expression of scorn (Pyne, 1982).

Supporters of burning did so for both philosophical and practical reasons--burning being the "Indian way" as well as expanding pasture and reducing fuels for forest fires. The detractors argued that small fires destroyed young trees, depleted soils, made the forest more susceptible to insects and disease, and were economically damaging. But the critical argument put forth by the opponents of burning was that it was inimical to the Progressive Spirit of Conservation. As a modern people, Americans should use the superior, scientific approaches of forest management that were now available to them, and which had not been available to the natives. Worse than being wrong, accepting native forest management methods would be primitive.

Bernhard Fernow, a Prussian-trained forester, thought fires were the ‘bane of American forests’ and dismissed their causes as a case of ‘bad habits and loose morals’. - Pyne (1995).

Through the early 20th century, the idea that fire was bad under all circumstances, and fire control must be based on suppression of all fires came became the dominant conservation ideology. After WWII the idea became stronger still, partially because of the availability of military equipment; but also due to the Cold War mentality. Just like Communism, the spread of fires simply couldn't be tolerated--and it was the duty of America to contain both "red" menaces (Pyne, 1982).


In the latter part of the 20th century, the ideas behind fire suppression once again began to change. The emphasis on "modern" methodologies began to fade, with a preference appearing for restoration of the "old forest" from pre-settler times. Research into the forest had begun to reveal the importance of fire in the natural setting, and that humans had used fire to manage the forest throughout history. Costs of fire suppression had risen dramatically, and the damage done to the forest by the equipment and the methods of fire suppression often exceeded that done by the fires.


Gradually the idea of fire suppression faded, to be replaced by a determination to allow fire to return to its natural role. Major fires in Yellowstone Park in 1988 brought about something of a reversal again in policy, but it was recognized that a century of fire suppression efforts had left the western forests in a dangerous state. Even though fire was to return to that natural cycle, the huge growth of underbrush has created a substantial risk of massive, out-of-control fires. This risk is an indicator of just how unhealthy fire suppression has made American forests.


By comparison, forests in Mexico, where there have been no fire-suppression efforts are far healthier. Fires are more common, but tend to be smaller, due to lack of fuel. 

Fire, water, and government know nothing of mercy. - Latin proverb

Default cascades as avalanches


Economic fluctuations have long been recognized as SOC phenomena. One type of fluctuation that has been recently posited is the "cascading cross default" in which the failure of one entity to repay its debts drives one (or more) of its creditors into bankruptcy, which in turn drives one or more of its creditors into bankruptcy, and so on.


Clearly these default cascades can be of nearly any size. A default may only affect the defaulting institution--or it may take down all institutions in a global collapse. As a conceptual model, the sandpile automaton of Bak et al. (1987) is a pretty good representation--the key difference being that each individual stack in the economic sandpile is actually connected to a large number of other stacks, some of which are (geographically) quite distant. For instance, the failure of Deutsche Bank would likely put stress on Citigroup. Would it cause it to fail? Perhaps. We would model this by assigning a probability of failure for Citigroup in the event of a default by DB. And we would have to do this for all relationships between the different banks.


But we need conditional probabilities--because it may be that DB's failure alone wouldn't topple Citigroup. But suppose it topples ING, and Credit Suisse, and Joe's Bank in Tacoma, and Fred's Bank in Springfield, and Tim's Bank in Akron, . . . and many others, all of whom owe money to Citigroup. Then it might fall. So apart from having tremendous interconnectivity, with each bank connected to many others, there is also tremendous density of those connections, all of which would appear to make the pile very unsteady. 


Instead of dropping grains of sand one at a time on the same spot, multitudes of debt bombs are dropped randomly on the pile of financial institutions, provoking episodic failures. What might we expect of their size distribution?

The experiment as I've described is too difficult to set up on my computer, mainly because I don't know how to establish the probabilities of failure for all of the various default chains that may exist. Furthermore, the political will to prevent financial contagion, although finite, is unmeasurable. Luckily we don't have to run the model, as it is playing out in real life.

Paper now primed to burn


We have lived through a long period of financial management, in which failing financial institutions have been propped up by emergency intervention (applied somewhat selectively). Defaults have not been permitted. The result has been a tremendous build-up of paper ripe for burning. Had the fires of default been allowed to burn freely in the past we may well have healthier financial institutions. Instead we find our banks loaded up with all kinds of flammable paper products; their basements stuffed with barrels of black powder. Trails of black powder run from bank to bank, and it's raining matches.

References

Bak, P., Tang, C., and Wiesenfield, K., 1987. Self-organized criticality: An explanation of 1/f noise. Physical Review Letters, 59: 381-384.


Pyne, S. J., 1982. Fire in America: A cultural history of wildland and rural fire (cycle of fire).