Power Laws | Kaelan

In a normal distribution, most values cluster near the average. In a power law distribution, a few extreme values dominate the total. The top 1% of cities hold more people than the bottom 50%. The top 1% of books sell more copies than the bottom 90%. The most prolific earthquake releases more energy than thousands of small ones combined.

The mathematical signature: values decrease as a power of their rank. The second-largest is half the first; the third is a third; the tenth is a tenth. Plot on log-log axes and you get a straight line. The distribution has no characteristic scale — it looks the same whether you zoom in or out.

Power laws break intuition trained on averages. In a normal world, the average is typical. In a power law world, the average is misleading — most observations fall far below it while a few tower above. Bill Gates walks into a bar and the average net worth becomes billions, but nobody in the room feels richer.

This is why fat tails matter for risk. Power law events don’t average out; they accumulate. One pandemic kills more than decades of flu seasons. One earthquake destroys more than centuries of tremors. Planning for the average leaves you defenseless against the extreme.

Power laws emerge from preferential attachment: the rich get richer, the connected get more connected, the visible get more attention. Success breeds success through feedback loops. First-mover advantages compound. network effects amplify.

The practical implication: in power law domains, being slightly better can mean being vastly more successful. Position matters more than small differences in quality. The gap between first and second is larger than between second and tenth. Winner-take-most markets reward strategy over incremental improvement.