petros, not at all thanks for asking a interesting question
(also, I have been reading your posts about your project in Athens
with great interest!)

this is basically a graph theory question. an easy way to calculate
this would be to empirically generate networks with certain properties
and then measure their size. This is fairly easy to do with some
computer programming know-how, which is easier to come by (in my
experince) than pure math know-how.

it's easy to generate a random graph (this is not what you are asking
for, but provides a starting point) by generating a set of nodes, then
choosing a number of edges (connections between nodes) and creating
random edges between nodes. (say, iterate over every pair of nodes,
and add and edge between them with some probability choosen before
hand)

This would generate a random graph, and there would be cases where you
have friends of friends, etc. now, you could make this more like a
social network by randomly choosing nodes, removing an edge, then
re-adding that edge to one of their friend's friends randomly. as you
keep doing this, you'll move the graph towards a social graph. but
also, formerly connect peers (or whole subgraphs) may become
disconnected from the graph. you could process the graph like this and
draw a graph of what size the largest connected subgraph is.

another way, which is possibly more intuitive, is you could start by
generating a random graph, and then have nodes "introduce friends"
randomly select a node, then randomly select two of it's friends, and
then make a new edge between them, until the proportion of shared
friends reaches your target.

while you are generating the graph, measure the proportion of mutual
friends, and then stop when you have the amount you are looking for.

another approach, is to generate an optimal graph, where you have the
maximum number of nodes which a certain graph. But I like the random
approach because surely a real human network is more random.

Dominic

On Fri, Jan 13, 2017 at 9:09 PM, Petros at Freelab <freelab@???> wrote:
> I apologise for the OT, but this is about quantitative modelling of small
> communities and it IS connected with hacking AND squatting so I hope someone
> would be interested in helping me out.
>
> ***
> tl;dr: I need a mathematical function to estimate a size of community, based
> on (actual) Dunbar number and the percentage of shared social contacts.
> ***
>
> My definition of small community is a community able to govern itself
> without institutional structures, based on natural mechanisms of social
> control and assessment. It means to me that the size of community must
> conform two principles:
>
> A number of direct social contacts (one direct friends, relatives or
> acquaintances) stays within the limit of Dunbar number (strict: 150,
> generous: 250)
> Throughout the community only two degrees of social separation exist. That
> means that for every member of community all other members are either one's
> direct social contacts or direct social contacts of his/her direct social
> contacts (“friends of a friend”).
>
> The effect of this is making all social interactions simple. I can identify
> everyone as a member of community. If I need an interaction, I can introduce
> myself, referring a common acquaintance. Most probably (gossiping is an
> important social function) we already know each other's reputation. In case
> of conflict, we can easily find one or more mediators. Etc., etc.
> A simple calculation tells us that absolute theoretical maximum size of such
> community lays between 150^2 and 250^2 (22 500 and 62 500 people,
> respectively). But this makes no practical sense, as it only happens if we
> have no common (shared) contacts. Thus, actual size of the community will
> depend of this very parameter: how many contacts (on average) we share with
> our direct ones. In extreme, again, it will be 100%, making the community
> size 152 or 252. But the real life spreads between extremes.
> Now, what I need from my illustrious friends (and other social contacts) are
> two things.
>
> Can anybody help me formulate a nice mathematical function, to estimate a
> size of community, based on the parameters listed above? Or perhaps (likely)
> such formula already exists, only I was unable to find it. Then, point me in
> the right direction, please.
> Can anybody help me establish what are empirical values of the shared
> contacts percentage in various communities and what influences that? Most
> probably it is known already, I am just unable to ask proper questions,
> while googling.
>
> Having it all together, I will be able to declare what actually is (in
> qualitative terms) a “community” I refer to so often in my musings on
> confederated system.
>
> Anybody, please?
>
> Best, Petros
>
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