## Network Laws Round Up

| August 21st, 2006**Sarnoff’s Law**

Sarnoff’s Law states that the value of a broadcast network is equal to the number of recievers (viewers/readers). Hence for a broadcast network of **n** users would have a value of **n:**

**n**

**Metcalfe’s Law**

Originally targeted for the Ethernet Metcalfe’s Law states that the value of a network is proportional to the square of the number of nodes (users) in the network. That is the value of a network having **n **users would be proportional to ** n^{2}**. Consider a network having 4 users:

As can be seen (in figure 1a) that each user can connect to every other user in the network which amounts to a total of n*(n-1) i.e. 4*(4-1) = 12 links

Removing the duplicate linkages it amounts to 4*(4-1) /2 = 6 as can be seen in figure 1b.

Ineffect it amounts to calculating the number of diagnols in the n-agon:

**Reed’s Law**

Reed’s Law states that the value of a network grows exponentially with the size of the network. It is based on the assertion that the value of a network is not measured by the number of users in it but rather by the number of sub-groups that can be formed by the users. Let us consider a network consisting of 3 users:

As it can be seen in the above figure that the number of possible sub-groups in a network of **n** users can be calculated by as **2 ^{n}**. Hence a network of 3 users would have 2

^{3}= 8 posiible sub-groups. Now this value can be broken as consisting of:

- sets with (number of users) > 1
- sets with (number of users) = 1
- empty set

Since the empty set and sets with only one user don’t count as a valid group the final estimation as per Reed’s Law turns out to be:

**2 ^{n} – n – 1**

**Refutation Of Metcalfe’s Law**

Real world communication networks, in general, grow faster than the linear growth (Sarnoff’s Law) but much slower than the quadratic growth (Metcalfe’s Law). Hence this law states that the value of a communication network of size **n** grows like:

**n log(n)**

The basis of this law is that Metcalfe’s and Reed’s Law are based on the false assumption that all connections or all groups are equally valuable.

The defect in this assumption was pointed out a century and a half ago by

Henry David Thoreauwhen he wrote:“We are in great haste to construct a magnetic telegraph from Maine to Texas; but Maine and Texas, it may be, have nothing important to communicate.”

**Zipf’s Law** states that if we order some large collection by size/popu;arity then the value of the k-th ranked item will be about 1/k of the first one. Hence by Zipf’s Law the value of a collection of n items is proportional to **log(n)**.

Now let us suppose that the incremental value that a person gets from other people being part of a network varies as Zipf’s Law predicts. Let’s further assume that for most people their most valuable communications are with friends and family, and the value of those communications is relatively ﬁxed – it is set by the medium and our makeup as social beings. Then each member of a network with n participants derives value proportional to

log(n), forn log(n)total value.

If we lose our fight against net neutrality, then this law would be more appropriate than Metcalfe’s Law, since then connections between certain nodes in the network would become more valuable than the others.

I wonder what law did Yahoo! and Microsoft use for evaluating the value of merging their IM networks. Any guesses?