Reed's law

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

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

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

Given a set A of N people, it has 2^N 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 2^N − N − 1 subsets remain, which is exponential, like 2^N.

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

Other analysts of network value functions, including Andrew Odlyzko and Eric S. Raymond, have argued that both Reed's Law and Metcalfe's Law 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.

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