What is Metcalfe's law?

A network's value grows as the square of the number of users — specifically n(n−1)/2 connectable pairs — so doubling users roughly quadruples value.

How do Sarnoff, Metcalfe and Reed's laws differ?

Sarnoff's law is linear (n) for broadcast networks, Metcalfe's is quadratic (n²) for pairwise networks, and Reed's is exponential (2^n−n−1) for group-forming networks.

Metcalfe's law

Add one user. Draw every new line.

A new user doesn't add one connection — they connect to everyone already here. That's why value snowballs while the user count barely moves.

Users · n

Pairs · n(n−1)/2

Highest law value

Number of users10

The three value laws

Sarnoff

broadcast · n

Metcalfe

pairwise · n(n−1)/2

Reed

group-forming · 2ⁿ−n−1

1,013

Tap a law to mute it on the plot. The graph on the left always draws the Metcalfe pairs.

Value vs users · log scalen = 10

Sarnoff n Metcalfe n² Reed 2ⁿ

What you're looking at

The left panel draws every pairwise connection in the network. Move the slider: each user you add must link to all the others, so the line count climbs as n(n−1)/2 — quadratically.

The plot uses a log vertical axis. On a linear axis the three laws would be unreadable; in log space they become three straight-ish lines fanning apart — slow Sarnoff, steeper Metcalfe, near-vertical Reed.

The three laws

Sarnoff (broadcast, n): radio, TV. Each user adds one constant unit of value — they only receive.

Metcalfe (pairwise, n²): telephone, email, fax. Value tracks the number of connectable pairs, which grows as n(n−1)/2.

Reed (group-forming, 2ⁿ): Slack, Facebook groups. Value counts the possible subgroups, of which there are 2ⁿ−n−1 non-trivial ones — exponential.

Why platforms snowball late

Below critical mass, n² is small and a network looks like a dud for years. Past it, value explodes faster than the user base — a 1.5× user base is a 2.25× Metcalfe value. The S-curve hides, then dominates; most networks die before reaching the bend.

Twitter at 1,000 users = 499,500 pairs. At 100,000 = ~5 billion pairs. The gap between "small forum" and "platform" is one decade of compounding the same simple rule.

Is it really n²?

Maybe not exactly. Briscoe, Odlyzko & Tilly argue that not all connections carry equal value — most pairs never interact — so real value grows closer to n·log n than n². The qualitative story survives: value still outpaces users. It just isn't as violent as the pure square.

Citations

Metcalfe, R. (2013). "Metcalfe's law after 40 years of Ethernet." Computer 46(12): 26–31. The original formulator's retrospective with empirical evidence from Facebook user-value data.
Briscoe, B., Odlyzko, A. & Tilly, B. (2006). "Metcalfe's law is wrong." IEEE Spectrum 43(7): 34–39. Argues O(n·log n) fits better than O(n²) once heterogeneity in connection-value is accounted for.
Reed, D.P. (1999). "That sneaky exponential — Beyond Metcalfe's law to the power of community building." Context Magazine Spring. Origin of Reed's Law (2ⁿ for group-forming networks).
Sarnoff, D. (mid-20th century, broadcast-era attribution). Linear value for broadcast networks where users are pure receivers.

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