Burn and Crash: percolation, Wall Street, Covid-19, forest fires, and mass extinctions
A number of years ago, my explorations into fractals led me to the amazing concepts of percolation theory and how they could be used as a heuristic tool to explore a whole range of phenomena. The basic concept of percolation theory is easy to explain, using the “forest fire” model. Imagine a square grid, some 100 cells on each side. Now randomly plant a tree somewhere in the grid. Plant another and another until about 10% of the grid cells have a tree on them. Next, send down a lighting bolt down onto the grid. If it hits a tree, the tree catches fire. The burning tree can ignite any tree in an adjoining cell (north, south, east, or west) and those trees can in turn ignite their neighbors. When only 10% of the cells are planted, most trees are isolated or are in small groups. Any fire quickly dies out. Let’s plant more trees. At 20% of the cells occupied, the fires are bigger but still isolated. The same is true at 30%, 40%, and 50%. The trees are in bigger, but still isolated groups. Fires, still started by a single lightning bolt, only burn a part of the grid. It is when the percentage of occupied cells starts to reach 60% that something remarkable happens. At about 59.28% of cells occupied by trees, there is about a 50% chance that the fire will spread from one side of the grid to other; it “percolates.” If one looks at the size of the largest connected cluster of trees on the maps, it jumps non-linearly upward at that value. Although we have not changed the local rule: “if you catch fire you burn your immediate neighbor,” the behavior of the entire system has changed. What was a small local fire becomes a system-wide conflagration beyond this critical value. We’ll add one more thing; areas that are burnt can regrow.
In an alternative formulation, every cell is initially occupied; imagine a newly planted forest. What changes over time is the probability of spread of the fire between trees. Think of it as the growth of canopies and the deposition of litter. The behavior is basically the same, once the probability of spread reaches about 50%, the fire can reach across the grid. The first form of the model is “site percolation,” the second is “bond percolation.” Many options exist, for example, changing the geometry of the grid, but the basic concepts are the same.
Many things strike me about these models. First, the behavior of the entire system cannot not be predicted from the local interactions. I use this in teaching as a simple example of “emergent properties.” Second, although the disturbance, the lightning strike, is always the same size, the size of the resulting fire depends on the structure of the system at the time the lightning strikes. To put it another way, the state of system determines how it responds to the disturbance. A system made of localized units prevents spread. Third, the system the amount of organization in the system, in turn, depends on how frequent the disturbances occur. Frequent lightning bolts will produce lots of local fires and the trees will never grow enough for a fire to percolate. If lightning bolts are infrequent, on the other hand, then the system builds up way beyond the critical point and the resulting fires are huge; the system resets to a much earlier point. Following this, there is a built-in lag between large fires. Enough trees must regrow for large clusters to exist.
But what really excites me is that this simple model and its behavior are highly adaptable. Replace fire with beetles, and you can model the spread of insects or of invasive species. Allow some trees to be resistant and others to be susceptible. Replace the grid with pores among sediment particles, and you can model the movement of water or oil. Get rid of the grid, and you end up with models for the spread of infectious diseases; there are a huge number of “epidemic models” for this purpose (including one that models a zombie outbreak). Or as I told a group of computer scientists at IBM, long before the development of the internet, the spread of computer viruses would not occur if people did not share floppy disks, creating an interconnected network.
My colleague Michael McKinney and I applied this model to what was once a major issue in paleontology; why are mass extinctions apparently periodic? In this case, we assumed that ecosystems begin in a disorganized, unconnected state and that they become more interdependent over time. Once they become highly organized, then what was once a small disturbance can spread and crash the whole system. The periodicity is produced by the lag time to rebuild connectivity. If the system is disorganized, even a hard whack will not spread, whereas if it is organized, even a small initial disturbance can be devastating.
The recognition that interconnectivity and interdependence can promote system fragility has been much on my mind lately. As is often pointed out, we now have a global economy that is highly interconnected. A drop in oil production due to a missile attack, an increase in tariffs, or a disease among workers in a manufacturing hub is not of local or regional concern; it can reverberate throughout the entire world economy (and hit my retirement account). No individual, region, or country is self-sufficient. This is both a strength and a source of potential widespread damage.
As I write this, I am “sheltering in place” in response to the Covid-19 outbreak. Humans live in large and interconnected social networks that promote the spread of diseases. We must break these bonds to stop the promulgation of the virus. To keep our society safe, we are returning to isolated bands. Keeping our “social distance” should drop the probability of spread below the critical threshold.
To all of you: stay safe, stay healthy, and maintain your social distance!
Note: an accessible guide to percolation theory is: Stauffer, D., and A. Aharony. 1994. Introduction to Percolation Theory. Taylor & Francis, London.
If you want to play with percolation and related models, such as agent models: NetLogo is fun and simple to use. https://ccl.northwestern.edu/netlogo/
And a recent summary I wrote:
Plotnick, R. E. 2017. Lattice models in ecology, paleontology, and geology. Pp. 83–94. In D. A. Budd, E. A. Hajek, and S. J. Purkis, eds. Autogenic Dynamics and Self-Organization in Sedimentary Systems SEPM Special Publication.
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