Power Laws, explained.
A power law describes a relationship in which one quantity scales as a fixed power of another. Certain power-law distributions produce strong concentration in a long tail.
Why it happens
Equal percentage changes matter more than equal absolute changes in a scaling relationship. In the rank-based experiment, a larger exponent gives the highest-ranked pages more traffic and leaves less for the rest.
A power relationship can allocate very different shares to different ranks. Here, the probability of visiting a page is proportional to 1 / rankᵃ. Increasing a makes the highest ranks more dominant.
At exponent zero, every page has the same probability. Random counts still vary. As the exponent increases, a small minority of ranks receives much of the traffic even before sampling variation is added.
Read the result
Compare the top-tenth traffic share with the average visits per page. Resampling changes the realized counts, while changing the exponent changes the underlying allocation probabilities.
A worked example
A site depends on a few popular pages
Imagine 100 pages receiving a fixed total number of visits.
An even allocation spreads attention broadly; a steep rank distribution makes a handful of pages responsible for much of the traffic.
The total may look healthy while the typical page receives little. Concentration also exposes dependence on the leaders.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The probabilities are normalized over a finite set of 10 to 100 ranked pages. One thousand independent visits are drawn from that distribution. Ranks do not change after a visit.
The top-tenth statistic compares observed traffic with its exact expected share. This is a rank-based, Zipf-style model, not a sample from an unbounded continuous Pareto distribution.
Where this idea is useful
An average can hide how strongly a total depends on a few contributors. Looking at concentration helps reveal that dependence.
A common misconception
“Any skewed chart proves a power law.”
Other distributions can look similar. Establishing a power law requires statistical comparison over an appropriate range, not just a straight-looking plot.
What this explanation leaves out
- The model imposes a power law; it does not show how one emerges or establish that real data follow it.
- All moments are finite because the number of pages is bounded. Claims about infinite variance do not apply to this finite model.
How is this different from the Pareto principle?
The Pareto principle is a rough concentration heuristic. A power law is a particular mathematical relationship. Neither guarantees that exactly 20% of cases produce 80% of the total.
Would you rather know the average contribution or how dependent the total is on its largest contributors?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Easley and Kleinberg discuss popularity, cumulative advantage, and power laws in Networks, Crowds, and Markets.