Systems and complexity
Nonlinearity
Cause and effect are not proportional: a push can do nothing or everything depending on where the system sits relative to a threshold.
The mathematics of dynamical systems; made a working idea for managers by system dynamics (Jay Forrester, Donella Meadows).
In a linear system doubling the input doubles the output and separate causes add up. In a nonlinear one the response follows a curve with thresholds, saturation and tipping points: below a threshold a change does nothing, above it the whole system moves. A drug dose, the load on a beam and the current in a power line all behave this way. Our intuitions are linear because daily life mostly sits on the gentle middle part of a curve, so we extrapolate from the slope we have seen and are surprised by the bend we have not.
The mechanism is usually feedback plus limits. In a network, when one element fails its load shifts to its neighbours, which can push them past their own thresholds, and a local fault becomes a system-wide one. That is why a cascade can take an hour to start and minutes to finish. In practice the rule is to ask for the shape of the curve, not the slope. A service that was fine at 80% utilisation tells you nothing about 95%, because queueing delay grows slowly and then explodes as utilisation nears 100%.
Nonlinearity cuts both ways: at the right point a small intervention has an outsized good effect, which is what Meadows means by a place to intervene. The common error is extrapolating a linear fit beyond the data it was fitted on; the 2003 task force found FirstEnergy's operators working for over an hour without alarms, unaware that lines had already tripped, and so treating the grid as if it were still in its normal range. The opposite error is calling everything a tipping point without naming the mechanism and the threshold, which turns a precise idea into a mood. The test of a real claim is that it says where the threshold is and what crosses it.
Example
On 14 August 2003 FirstEnergy's control-room alarm software in Ohio failed at 14:14 EDT and nobody noticed. At 15:05:41 its Harding-Chamberlin 345 kV line sagged into a tree and tripped while carrying only 44% of its rating; as load shifted, the Hanna-Juniper line tripped at 15:32 and Star-South Canton at 15:41 the same way. When the Sammis-Star line tripped at 16:05:57 the cascade began, and within seven minutes, by 16:13, more than 508 generating units at 265 power plants had been lost. The US-Canada task force report of 2004 put the blackout at an estimated 50 million people and 61,800 megawatts of load.
Where it applies
Load-test a service at 95% utilisation, not 50%, and look for the knee in the latency curve. In machine learning, model quality against data or compute follows curves rather than straight lines, so a result from a small run rarely scales the way a linear fit predicts.
Think about it
Where are you extrapolating from the linear part of a curve, and what sits past the threshold you have never tested?
Applied it to something real? Log it.
Test yourselfThree power lines fail over an hour, then the whole grid goes in seven minutes. What property of the system does that show?
Nonlinearity: each failure shifted load onto neighbours until one trip crossed a threshold and the cascade fed itself. The effect was not proportional to the cause.
Related models
- Feedback loopsA system's output feeds back into its input: balancing loops hold a level steady, reinforcing loops make change grow on itself.
- EmergenceSimple local rules among many parts produce patterns at the level of the whole that no part contains or intends.
- Margin of safetyLeave a wide gap between what you expect to need and what you build or pay, so that your errors and bad luck do not sink you.
- Probabilistic thinkingThink in odds rather than certainties: estimate how likely each outcome is, and judge a decision by those odds rather than by the single result that happened.
- Exponential versus linear growthLinear growth adds the same amount each step; exponential growth multiplies by the same factor, and the two look alike only at the start.