Distributions

Rich get richer

Two worlds. Same songs. Same listeners. One knob difference.
Result: one world is fair. The other one has a runaway hit.

In 2006 Salganik, Dodds & Watts ran a real experiment called MusicLab. Thousands of people were split into parallel "worlds" listening to the same set of unknown songs. In one world, listeners couldn't see anyone else's download counts. In another, they could. Same music, same appeal, one difference in what listeners knew. The download distributions came out radically different — and unpredictable. Let's re-run it below.

The two-worlds experiment · social influence vs independence

Left world · independent. Each new listener picks a song using only its inherent appeal (hidden — all songs are of similar quality). · Right world · social influence. Each listener also considers how many others have already picked it. Popular = more likely to be picked. Rich get richer.

Hit reset a few times. Each fresh run in the left world stays fairly even. Each fresh run in the right world produces a runaway hit — but which song wins changes every time. That's Salganik & Watts's headline finding: with social influence, popularity is not just unequal, it's unpredictable. A song that flops in one alternate universe is the #1 in another. Quality alone doesn't decide.

The signature: a straight line on log-log

The right world's histogram — most songs get a handful, a couple get hundreds — is a power law. Its fingerprint is a straight line on a log-log plot of "how many songs have this many listeners?" That line shows up in city sizes, word frequencies, wealth, earthquake magnitudes, and Wikipedia edits. Different mechanisms, same shape.

Log-log degree distribution · the power-law fingerprint

Network size2000 Edges per new node (m)2

Fitted exponent γ̂

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theory ≈ 3

How unequal is it, really?

A Lorenz curve answers the question in one picture. On the x-axis: nodes ranked from least to most connected. On the y-axis: cumulative share of all links. A perfectly equal world would be a straight diagonal. Reality — anything preferential-attachment produces — bulges downward.

Lorenz curve · concentration of degree

Gini coefficient

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0 = equal · 1 = one hub only

Top 20% own

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the "80/20 rule" if ≥ 80%

Where power laws appear. City sizes, earthquake magnitudes, casualties in wars, book and music sales, website in-links, word frequencies (Zipf), wealth, firm sizes, forest-fire areas. Different generating mechanisms, same shape — meaning fitting a straight line to log-log doesn't identify the cause.

Related models

🔔 Normal / CLT — the bell-curve world. Same data-generating ingredients (summing vs multiplying) flip the tail from thin to heavy.
🕸️ Networks — preferential attachment produces scale-free networks: the same mechanism, viewed as a graph instead of a distribution.
🏺 Path dependence — Pólya's urn and preferential attachment share the "small random early moves lock in big late outcomes" shape.

← Normal / CLT Networks →