The Math Beneath

On the surface, it doesn’t seem that financial modeling has much in common with climate science, ecology, or neuroscience. But in fact these fields are grappling with similar mathematical problems: how to map nonlinear, deeply interconnected systems and anticipate systemwide collapse, notes George Sugihara, a theoretical biologist at the Scripps Institution of Oceanography in San Diego.

For financial modelers, the challenge is to predict crashes. An investment banker looking at one portfolio will not be able to see the systemic factors that could lead to a meltdown. Likewise, a marine scientist trying to protect a species of fish will not be able to account for all of the variables in the system that affect the survival of that particular fish without looking beyond that one species. In the field of climate science, linear models cannot predict what we know historically to be true: that climate change can be rapid and extreme. The existing models are all very good for painting a picture of a complex system at a specific point in time, but they do not have the ability to explain “jumps in variability,” or what mathematicians call heteroscedasticity.   Complexity theorists—the mathematicians who explore these sorts of systems—are beginning to pinpoint some early warning signs of systemic collapse. One is that as systems get closer to meltdown, they become slower to respond to external stimuli. Another is that pulses occurring in neighboring parts of the web become synchronized. For example, nearby brain cells fire in unison in the lead-up to an epileptic seizure.   A similar pattern emerged before the recent financial crash. Over time, financial institutions’ investment holdings became more alike (the perverse result of each institution independently pursuing extreme diversification) and began to respond to changes in the market nearly simultaneously. When large financial institutions such as Lehman Brothers fell apart, the fallout was not unlike what happens in an ecosystem in which many animals rely for sustenance on one large animal species that suddenly dies out.   Heed Einstein, Sugihara advises: “Everything should be made as simple as possible, but not simpler.”