Reed's law

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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

• 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.

Derivation of the number of possible subgroups

Given a set A which represents a group of people, and whose members are persons, then the number of people in the group is the cardinality of set A.

The set of all subsets of A is the power set of A, denoted as [ \mathcal (A) ]:

[ \mathcal(A) = \ ].

It is known in set theory that the cardinality of [ \mathcal(A) ] is equal to 2 to the power of the cardinality of A, i.e.

[ \mbox \, \mathcal(A) = 2^ \, A} ].

This is not difficult to see, since we can form each possible subgroup by simply choosing for each element of A one of two possibilities: whether to include that element, or not.

However, the empty set [ \emptyset ] belongs to the power set [ \mathcal(A) ] but is not a group of people; hence we must subtract it out:

[ \mbox \, \left( \mathcal(A) - \emptyset \right) = 2^N - 1 ],

where [ N = \mbox \, A ].

Further, any members of [ \mathcal(A) ] which are singletons are not considered "groups of people". Since each individual in a group can form a singleton, then the number of singletons in A is equal to the cardinality of A:

[ \mbox \(A) \wedge \mbox \, C = 1 \} = N, ]

[ \mbox \, \left( \mathcal(A) - \emptyset - \(A) \wedge \mbox \, C = 1 \} \right) = 2^N - N - 1. ]

Notice that the function [ 2^N - N - 1 ] is exponential, in proportion to [ 2^N ].

Quote

From David P. Reed's, "The Law of the Pack":

"[E]ven Metcalfe's Law understates the value created by a group-forming network 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."

See also

• Coase's Penguin
• social capital
• Metcalfe's law
• McCandlish's Foil to Reed and Metcalfe (humor)
• Content is Not King
• Observations named after people

External links

• [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.]

 

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