Reed's law

From Academic Kids

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.

The reason for this is that the number of possible sub-groups of network participants is <math>2^N - N - 1 \, <math>, where <math>N<math> is the number of participants. This grows much more rapidly than either

  • the number of participants, <math>N<math>, or
  • the number of possible pair connections, <math>N (N - 1) / 2\,<math> (which follows Metcalfe's law)

so that even if the utility of groups being 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 <math> \mathcal{P} (A) <math>:

<math> \mathcal{P}(A) = \{B : B \subseteq A\} <math>.

It is known in set theory that the cardinality of <math> \mathcal{P}(A) <math> is equal to 2 to the power of the cardinality of A, i.e.

<math> \mbox{card} \, \mathcal{P}(A) = 2^{\mbox{card} \, A} <math>.

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 <math> \emptyset <math> belongs to the power set <math> \mathcal{P}(A) <math> but is not a group of people; hence we must subtract it out:

<math> \mbox{card} \, \left( \mathcal{P}(A) - \emptyset \right) = 2^N - 1 <math>,

where <math> N = \mbox{card} \, A <math>.

Further, any members of <math> \mathcal{P}(A) <math> 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:

<math> \mbox{card} \{C : C \in \mathcal{P}(A) \wedge \mbox{card} \, C = 1 \} = N, <math>
<math> \mbox{card} \, \left( \mathcal{P}(A) - \emptyset - \{C : C \in \mathcal{P}(A) \wedge \mbox{card} \, C = 1 \} \right) = 2^N - N - 1. <math>

But notice that — using Big O notation — the function <math> 2^N - N - 1 \, <math> is <math> O(2^N) \, <math> as <math> N \rightarrow \infty \, <math>, so that it is exponential.

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