Reed's law is the assertion ofDavid P. Reed that theutility of largenetworks, particularlysocial networks, canscale 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, whereN 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 followsMetcalfe's law).

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

Given asetA ofN 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 ofA one of two possibilities: whether to include that element, or not.

However, this includes the (one) empty set, andNsingletons, 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):

"[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 withn 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 generatenetwork effects that dominate the overall economics of the system.^[2]

Other analysts of network value functions, includingAndrew 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 aroundDunbar'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.