Definitions

# Reed's law

Reed's law is the assertion of David P. Reed that the utility of large Network, 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

• the number of participants, $N$, or
• the number of possible pair connections, $frac\left\{N\left(N-1\right)\right\}\left\{2\right\}$ (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.

## Derivation

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

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."