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 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 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.
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."
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.
- Andrew Odlyzko's "Content is Not King"
- Beckstrom's law
- Coase's penguin
- List of eponymous laws
- Metcalfe's law
- Six Degrees of Kevin Bacon
- Sarnoff's law
- Social capital
- That Sneaky Exponential—Beyond Metcalfe's Law to the Power of Community Building
- Weapon of Math Destruction: A simple formula explains why the Internet is wreaking havoc on business models.
- KK-law for Group Forming Services, XVth International Symposium on Services and Local Access, Edinburgh, March 2004, presents an alternative way to model the effect of social networks.