Question
What advantages do restricted membership groups have over public open groups?
Answer
I like an idea I inferred from Geoffrey West's (Santa Fe researcher) recent work on cities vs. corporations: the more open a group is, the more likely it is to be long-lived or even effectively eternal. The more closed it is, the more it will behave like a mortal being, with a definite birth, growth, maturity and decline lifecycle.
Let's say openness is a number from 0 to 1, with 1 being completely open. Then I'd say the required openness, [math]P(G)[/math] for a group [math]G[/math] is likely governed by a constitutive law that looks something like:
[math] P(G)=\frac{S\times L}{S_{\max}\times L_{\max}} [/math]
where [math]S[/math] is the size of the group, [math]S_{\max}[/math] is the size of the population affected by the actions of the group, [math]L[/math] is the design lifespan of the group, and [math]L_{\max}[/math] is the known maximum lifespan among historical instances of that group.
This is a zeroth-order formula. You probably want to fuzzify the notion of membership in the group and impacted group for starters. You'd also need a [0,1] definition of "openness." It would also be fun to model the social graph within this formula, via notions of proxying and relatedness of membership (is a husband represented in a group if a wife joins?)
The nice thing about this formula is that it can produce an answer greater than 1 (if you take into account, for instance, some proxy of the opinions of future generations, and the possibility that you could achieve a level of openness that has no precedent in history...). This turns openness into a concept that is constantly being redefined and renormalized to [0,1].
Okay, I admit that half of the reason I wrote the answer this way is that I simply get a kick out of typsetting Latex for no good reason. There's nothing here that actually needs a math formula.
Let's say openness is a number from 0 to 1, with 1 being completely open. Then I'd say the required openness, [math]P(G)[/math] for a group [math]G[/math] is likely governed by a constitutive law that looks something like:
[math] P(G)=\frac{S\times L}{S_{\max}\times L_{\max}} [/math]
where [math]S[/math] is the size of the group, [math]S_{\max}[/math] is the size of the population affected by the actions of the group, [math]L[/math] is the design lifespan of the group, and [math]L_{\max}[/math] is the known maximum lifespan among historical instances of that group.
This is a zeroth-order formula. You probably want to fuzzify the notion of membership in the group and impacted group for starters. You'd also need a [0,1] definition of "openness." It would also be fun to model the social graph within this formula, via notions of proxying and relatedness of membership (is a husband represented in a group if a wife joins?)
The nice thing about this formula is that it can produce an answer greater than 1 (if you take into account, for instance, some proxy of the opinions of future generations, and the possibility that you could achieve a level of openness that has no precedent in history...). This turns openness into a concept that is constantly being redefined and renormalized to [0,1].
Okay, I admit that half of the reason I wrote the answer this way is that I simply get a kick out of typsetting Latex for no good reason. There's nothing here that actually needs a math formula.