Metcalfe vs Zipf
How network value really scales—not quadratically like you think.
Tiny Summary
Metcalfe's Law (n²) overstates network value. Zipf's Law (n·log n) is more accurate—connections aren't equally valuable, so growth is logarithmic, not quadratic.
Metcalfe's Law
Formula: V = n²
Claim: Network value grows with square of users
100 users → Value = 10,000
1,000 users → Value = 1,000,000 (100x!)
The Problem
Assumes all connections equally valuable:
- 1st friend: Very valuable
- 10th friend: Still valuable
- 100th friend: Somewhat valuable
- 1,000th friend: Barely know them
Reality: Diminishing returns, not all connections matter, active ≠ total users
Zipf's Law
Formula: V = n · log(n)
Claim: Network value grows logarithmically (power law distribution, 80/20 applies)
100 users → Value ≈ 460
1,000 users → Value ≈ 6,900 (15x, not 100x)
The Comparison
Users | Metcalfe (n²) | Zipf (n·log n) | Ratio
------|---------------|----------------|-------
100 | 10,000 | 461 | 21.7x
1,000 | 1,000,000 | 6,908 | 144.8x
10,000| 100,000,000 | 92,103 | 1,086x
As networks grow, Metcalfe increasingly overstates value.
Real-World Evidence
Facebook/Social Networks: Actual growth logarithmic (Zipf), not quadratic. You don't interact with all 500 friends—90% of interactions with top 10-20.
Tencent Study (2015): WeChat and QQ data validated n·log(n), not n²
Why It Matters
Valuations: Don't say "1M users = Metcalfe value X". Say "100k active users with engagement Y"
Projections: 10x users → 30-40x value (logarithmic), not 100x
Focus: Engagement and retention (power law users), not pure acquisition
When Each Applies
Metcalfe works: Small fully-connected networks (Slack team < 50 people, family chat), high-engagement platforms (trading, marketplaces)
Zipf wins: Large social networks (most are lurkers), communication platforms (message same 10 people 90% of time), content platforms (watch same creators repeatedly)
Key Insights
Metcalfe oversells—assumes all connections equal, leads to bubble valuations. Zipf is realistic—matches empirical data, accounts for power laws. 1,000 active users beats 10,000 inactive. Quality of network beats size of network. Focus on retention and engagement over acquisition.
Use the simulation to see how the models diverge!