# Reed's law

Reed's law is the assertion of David P. Reed that the utility of large networks, particularly social networks, can scale exponentially with the size of the network.[1]

The reason for this is that the number of possible sub-groups of network participants is 2N − N − 1, where N is the number of participants. This grows much more rapidly than either

• the number of participants, N, or
• the number of possible pair connections, N(N − 1)/2 (which follows Metcalfe's law).

so that even if the utility of groups available to be joined is very small on a per-group basis, eventually the network effect of potential group membership can dominate the overall economics of the system.

## Derivation

Given a set A of N people, it has 2N possible subsets. This is not difficult to see, since we can form each possible subset by simply choosing for each element of A one of two possibilities: whether to include that element, or not.

However, this includes the (one) empty set, and N singletons, which are not properly subgroups. So 2N − N − 1 subsets remain, which is exponential, like 2N.

## Quote

From David P. Reed's, "The Law of the Pack" (Harvard Business Review, February 2001, pp 23–4):

"[E]ven Metcalfe's law understates the value created by a group-forming network [GFN] as it grows. Let's say you have a GFN with n members. If you add up all the potential two-person groups, three-person groups, and so on that those members could form, the number of possible groups equals 2n. So the value of a GFN increases exponentially, in proportion to 2n. I call that Reed's Law. And its implications are profound."

Reed's Law is often mentioned when explaining competitive dynamics of internet platforms. As the law states that a network becomes more valuable when people can easily form subgroups to collaborate, while this value increases exponentially with the number of connections, business platform that reaches a sufficient number of members can generate network effects that dominate the overall economics of the system.[2]

## Criticism

Other analysts of network value functions, including Andrew Odlyzko, have argued that both Reed's Law and Metcalfe's Law [3] overstate network value because they fail to account for the restrictive impact of human cognitive limits on network formation. According to this argument, the research around Dunbar's number implies a limit on the number of inbound and outbound connections a human in a group-forming network can manage, so that the actual maximum-value structure is much sparser than the set-of-subsets measured by Reed's law or the complete graph measured by Metcalfe's law.