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 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 subgroupsEdit
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.
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 belongs to the power set but is not a group of people; hence we must subtract it out:
Further, 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:
Notice that the function is exponential, in proportion to .
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 Edit
- 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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