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The reason for this is that the number of possible sub-groups of network participants is , where is the number of participants. This grows much more rapidly than either

• the number of participants, , or
• the number of possible pair connections, (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.

1 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 :

.

It is known in set theory that the cardinality of is equal to 2 to the power of the cardinality of A, i.e.

.

However, A itself belongs to its own power set but if A is considered as a group of people, then A is not a proper "subgroup" of itself:

,

where .

Then, any members of 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:

But notice that — using Big O notation — the function is as , so that it is exponential.

2 See also

• Coase's Penguin
• social capital
• Metcalfe's law
• Content is Not King
• List of eponymous lawsThe list of eponymous laws provides links to articles on laws, adages, and other succinct observations or predictions named after a person. In some cases the person named has coined the law such as Parkinson's law. In others, the work or publications of t

3 External links

• That Sneaky Exponential—Beyond Metcalfe's Law to the Power of Community Building