From Stones to Sand: Self-Organized Criticality
Nothing is built on stone; all is built in sand. But we must build as if the sand were stone.
—Jorge Luis Borges
Here we begin with a pile of sand and, alas, end with one. We often observe that complex systems, after having developed a beautiful and seemingly robust structure, can collapse in an instant. Consider your body, a collection of billions of cells, each interacting and forming a recognizable and vital you. Yet all of those interactions, all that you are and could be, can cease in only a few minutes if, say, you experience a misplaced shock to your heart. Or consider a civilization such as the Maya in their Classic period, in which a vibrant Mesoamerican culture suddenly falls apart. Is there something innate about complex systems that demands an inescapable vulnerability to collapse?
To explore this question, let’s randomly sprinkle grains of sand on top of an empty table. At first, as the grains fall, they stay where they land. With time, an occasional grain lands on top of another grain, and as long as the new height is not much higher than that of the surrounding grains, it will balance. As the sand continues to pile up, we eventually reach a point where a grain can no longer balance where it falls, and a little avalanche ensues as the grain tumbles onto its neighbor. With few grains on the table, such tumbles result in a slight displacement of the ever-growing pile. However, as the sand continues to mound on the table, tumbles begin to cause imbalances at neighboring locations, resulting in new tumbles and a larger avalanche, perhaps even to the point where some sand falls off the edge of the table.
How such sand piles behave forms the core of a model of self-organized criticality developed by the physicist Per Bak. Sometimes a falling grain has little impact other than to add itself to the spot where it landed. At other times the grain begins an avalanche that triggers a chain reaction of additional grains tumbling across the pile. Indeed, avalanches of all possible sizes, following a well-specified probability distribution (yet another power law), characterize this system’s behavior (see Figure 11.1).
At any given time during our sand-pouring experiment, we could pause and take stock of the conditions at any location on the table. At every location, the pile is either subcritical (that is, adding a grain will just increase its height by one) or critical (that is, it’s teetering in such a way that the addition of a single grain of sand will cause it to tumble onto a neighboring spot). Every grain of sand we add, every tumble that occurs, is continually pushing the system toward a critical state. At times, large swaths of the pile are poised so that the addition of a single grain of sand will cause an avalanche across the entire area. After that avalanche has devastated the pile, the system has relaxed enough so that new additions of sand either stay where they land or result in only small, localized avalanches that are quickly absorbed by subcritical neighbors. Overall, we observe long periods of relatively localized turmoil that increasingly drive the system toward widespread criticality, setting the stage for even a small event to trigger another widespread avalanche.
There is a relentless logic that drives such systems. Above, we assumed a simple physics with nicely behaved grains of sand that topple due to gravity when they pile too high. Even if we modify the physics by making the grains more irregular or by altering the force of gravity, similar behavior emerges. Under these new conditions, the system is still driven toward critical states. So whether we experiment with beach sand here on earth or dust on the moon, the self-organized criticality of the system remains a fundamental, emergent feature.
While the sand pile is dominated by simple physics, other systems may be driven by other mechanisms. For example, criticality in social systems might depend on features such as laws and regulations or financial risk. Laws and regulations may have little impact on social behavior at times. But as the circumstances of agents change, policies begin to bind, forcing agents into critical states where even small events can trigger large responses. Thus, we might see segments of society rise up against the government’s taxation and fiscal policy, perhaps at first just forming local Tea Party–like movements, but on occasion triggering widespread social revolts. Or consider the banking and investment system, where various institutions try to maximize their returns by leveraging their assets and taking on risk. Over time, these systems can enter critical regimes where even small changes, perhaps the inability of one bank to repay a single loan, can result in a large avalanche as failure begets failure.
While in physical systems the drivers of criticality (such as gravity) are exogenous, in social systems they are often endogenous. Social elements such as tax rates and the amount of leverage banks are allowed to exercise are under the control of the government, typically through some political process. Political actors often have incentives to alter such policies in ways that might change the key determinants of criticality.
Consider a Classic-period Mayan city. The city proper is surrounded by farmers who must pay a tax to the government, either by turning over a share of their crops or by providing labor. In return, the farmers receive services from the city, such as protection, governance, and some insurance in case their crops fail. At low tax rates, the farmers are happy, because the amount they pay in taxes is more than compensated for by the services they receive. As the taxes are raised, the farmers become increasingly disgruntled with the trade-off they must make. At some point, things may become so bad that a farmer might rebel or move elsewhere.
Suppose our Mayan government, like most governments, prefers more revenue rather than less, perhaps because there is always a demand to build more elaborate temples. As the government raises tax rates, it starts to push the system closer to criticality. Every farmer is continually making a choice, weighing the benefits of staying at his current location against the tax he must pay. He’ll consider his investments in improving the fields, his network of friends, his ancestral ties to the place, and so on. As the tax rate rises, the imbalance between the benefits of staying and the costs of leaving lessens, and the farmer is pushed closer to a critical state where even a small change—some bad weather or the loss of a cooperative neighbor, let alone a new government demand—could cause the farmer to up and leave.
If a farmer decides to leave, we see impacts that resemble those in our sand pile. On one hand, the farmer’s field, now fallow and needing minimal investment to put it into production, might simply be taken over by someone else. Here, the departing farmer is like a grain of sand forming a subcritical hole in the pile. Alternatively, when the farmer leaves he might trigger his neighbors to leave as well. After all, the neighbors lost an important social connection who provided friendship and cooperation, and who by moving lessened the taboo against relocating the bones of one’s ancestors. This latter situation is much like a grain of sand surrounded by other grains all in a critical state.
Unlike physical systems, social systems are likely to embody additional endogenous forces that could accelerate their criticality. For example, the immediate loss of production from the relocating Mayan farmer may force the government to increase its taxation on the remaining farmers. This will cause a further increase in system-wide criticality. Indeed, such endogenous drives toward increased criticality may be a natural outcome of social governance, as governments in pursuit of their goals tend to push citizens toward action. Once the system becomes critical, even trivial external events or policy changes can provoke system-wide reactions.
The idea of self-organized criticality may provide some needed insight into social phenomena involving rapid collapse and change. The rapid abandonment of Mayan cities in the Classic period could have been presaged by years of social policy that forced the system into a critical state. Once the society was in this state, the dynamics of the sand pile took over. Any social system is continually perturbed by seemingly insignificant events such as bouts of bad weather, missteps by the ruler, and so on. These perturbations usually have few noticeable consequences. Perhaps, on occasion, a farmer and maybe a neighbor or two decide to leave and set up operations elsewhere, but nothing much more than that. Yet such actions and reactions slowly flow through the system and inexorably drive it to a critical state. Once there, a seemingly minor affront to the system can trigger a large-scale avalanche.
On December 17, 2010, a Tunisian street vendor, Mohamed Bouazizi, set himself on fire to protest years of harassment by authorities. The event that triggered his protest was a municipal official publicly humiliating him by confiscating the scale he used to weigh his produce. Bouazizi tried to complain to the governor, but the governor refused to see him. This led him to the act that would eventually take his life.
Thus began the Arab Spring, where the confiscation of a vendor’s scale in a rural Tunisian town started a wave of unrest that rippled outward from Tunisia into Algeria, Lebanon, Jordan, Mauritania, Sudan, Oman, Saudi Arabia, Egypt, Yemen, Iraq, Bahrain, Libya, Kuwait, Morocco, Western Sahara, Syria, and Israel’s border towns. The outcome to date is a number of revolutions resulting in dramatic changes in governments, harsh crackdowns, and diplomatic maneuvering. The full impact of these events on the course of world history will likely be significant, but it’s hard to fathom at this stage.
One can easily postulate forces, such as unhappy citizens or the dictates of an autocratic ruler, that could force a society into a critical state. Moreover, when one citizen is pushed so far that he decides to protest, this increases the likelihood that those nearby might take up the protest as well. Protests in various guises had been occurring in these countries for some time, but most of these efforts were quite localized. Nevertheless, they had been quietly driving the system to a more critical state. Once the system entered such a state, even an inconsequential act could trigger large-scale change, the consequences of which we are only starting to grasp. While such a hypothesis is speculative, one could test it by looking for the signatures of growing criticality in the various data feeds, such as Twitter, that may well have both captured and contributed to these events.
Self-organized criticality is an interesting form of complexity where small pieces of the system interact locally with one another, mediated by a very simple rule governing change. Over time, the system abstracts itself away from the particular local rule, and its global behavior is dominated by a characteristic pattern of avalanches at all scales. Most of these avalanches are small, but on rare occasions one encompasses the entire system. When global events occur, we want to invoke global causes. But the lesson from self-organized criticality is that there are forces underlying systems such that even small events, normally inconsequential, can have huge impacts.
At the slightest touch, our world can go from stones to sand.