# 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 2^{N} − *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.

## Contents

## Derivation[edit]

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}.

## Quote[edit]

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 2^{n}. So the value of a GFN increases exponentially, in proportion to 2^{n}. I call that Reed's Law. And its implications are profound."

## Business implications[edit]

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[edit]

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.

## See also[edit]

- 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

## References[edit]

**^**Hogg, Scott (October 5, 2013). "Understand and Obey the Laws of Networking: Ignorance of the laws of networking is no excuse".*Network World*. Retrieved November 2, 2017.**^**Heckart, Christine. "The network effect on wealth creation".*Network World*. Retrieved 2017-11-07.**^**"Metcalfe's Law is Wrong".*IEEE Spectrum: Technology, Engineering, and Science News*. Retrieved 2017-11-10.

## External links[edit]

- 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.