The math in the article: Metcalfe's law is wrong

Question

In this article: Metcalfe's law is wrong

There is a calculation:

Imagine a network of 100 000 members that we know brings in $1 million.

So if the network doubles its membership to 200 000

Metcalfe’s law says its value grows by (200 000^2 /100 000^2 ) times, quadrupling to $4 million

Whereas the n*log(n) law says its value grows by 200 000*log(200 000)/100 000*log(100 000) times to only $2.1 million.

My questions are:

• Why grows by (200 000^2 /100 000^2) times? The Metcalfe’s Law is as simple as n^2, how come there is a division ?
• How does the $4 million been calculated?

Answer 1

Why grows by (200 000^2 /100 000^2) times? The Metcalfe’s Law is as simple as n^2, how come there is a division ?

Because Metcalfe's law is used for calculating the proportional value of a network based on the "number of connections." In the article they are calculating the percentage of growth. So you take the new value and divide it by the old value.

How does the $4 million been calculated?

Because 200,000²/100,000² gives you an answer of 400% or 4 times the original value. 4 * 1 = 4

Answer 2

If I $\times2$ the amount of users, I expect to $\times4$ the revenue.

$4=2^2$.

In this case, $\displaystyle \frac{2 \,\,\text{million}}{1 \,\,\text{million}}=2$.

The division is to find the growth in membership. Square the membership growth to get revenue growth.