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July 11th, 2001 (02:25 am)

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\begin{document}
\begin{center}
Social Incidence Matrixes
\end{center}

\begin{quote}
So, you're caught in this situation: you've got many friends
and your social situation has rapidly erupted into a highly
complex and somewhat impermeable beast that you've got difficulty
sorting out. Enter the \textit{social incidence matrix}. Here's

\begin{tabular}{c|c|c|c|c|c|c|c|c|}\hline
{} & A & B & C & D & E & F & G & H \\
A  & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\
B  & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\
C  & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
D  & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
E  & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 \\
F  & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\
G  & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\
H  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & o \\
\end{tabular}

So, for friends {A,B,C,D,E,F,G,H}, we have a social
incidence matrix. I can bitch about one of my friends
to another friend in a given row if the value at the
position (row,column) is one. Likewise, for some
reason, those who I can't bitch about a friend
in a given column to a given row, have a value of zero.

Here's a formal definition. Choose a set of your friends
$F={f_{1},\ldots,f_{n}}$. Then $\forall i \ni 1 \leq i \leq n$, and
$\forall j \ni 1 \leq j \leq n$, define $a_{i,j}=1$ if and only if
you are able to discuss/bitch/complain about $f_{j}$ to $f_{i}$.
The entire matrix, $A$, gives you cute info.

\begin{tabular}{c|c|c|c|c|c|c|c|c|c|}\hline
{} & A & B & C & D & E & F & G & H & affinity\\
A  & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 3\\
B  & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 3\\
C  & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2\\
D  & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 7\\
E  & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 7\\
F  & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 7\\
G  & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 4\\
H  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
alienation & 6 & 5 & 5 & 2 & 2 & 2 & 4 & 7 \\
\end{tabular}

So for $f_{k}$, the \textit{affinity} to be

$\sum_{u=1}^{n} a_{k,n}$

And also, define the \textit{alienation}

$\sum_{u=1}^{n} a_{n,k}$

I think that the affinity and alienation are related: if you're
completely willing to talk bitch to most people about a given
persons's problems, then, you really don't expect to risking
it because they've completely alienated themselves from you
and everyone else. Also, you're usually not comfortable
so $\delta_{ij}a_{ij}=0$ is an assumption. If the determinant
of the social incidence matrix is singular, then there's
at least one row or column that has all the same entries.
The usual implication of a singular social incidence matrix
is that there exists two rows or columns that are linear
multiples of each other (which really won't happen), or
that an entire row or column is zeroed. The second
is altogether more likely.

Rows are zeroed if and only
if that person has entirely alienated themselves from you.
Columns are zeroed when your relationship is intensely private.
In both cases, the determinant of the social incidence
matrix will be singular. Both situations resemble each
other in that they both represent a restriction of
information passing.

\textit{Correction: a matrix with a single column or row
containing only zeros will have a determinant of zero. Therefore
if include an alienated person, you would instantly give the matrix
a zero determinant. If two people have precisely the same distributions
of whom you can and can't talk to issues about, the determinant is also
zero. Since column and row order doesn't matter, the absolute value
is probably more telling than the sign}

Alienated people and intensely private
relationships are circumstances in which communication
becomes extremely limited (alienated people aren't born
alienated, and private relationships are private, you never
hear about them). I suggest to you, dear social complexity
boatswain, that you not insert these sorts of people
in your incidence matrixes. The determinant would
also be singular if you could talk about everything
to everyone, or nothing to nobody (that is, if you had
alienated everyone, your matrix would be all zeroes).

Assuming that you haven't alienated everyone you know,
and that you don't live in a bizarre environment where
everyone speaks their mind and freely and equally bitches
about people to themselves in a semi-serious manner, then,
the next question is, what does that determinant mean?

If you were to remove the column and row for H on the
above matrix, then $|det(A)|=-2.0$.

\textit{Mapping determinants of social incidence matrices
onto meaningful sorts of things} A determinant of zero
means either 1. There's someone in the mix who completely
alienates you and everyone else. It's best to remove this person., 2.
There's a person whom you've got an intensely private relationship that
you can't mention to anyone (this is effective alienation). 3. There
are two people you know whom you can discuss issues concerning the
exact same set of people. Why haven't you introduced them to each other?
and 4. There are two people for which you have the same set of
trepidations and openness talking with about other people -- you should
introduce them to each other. A determinant of zero is an instant
sign that there is either first order (1,2) or second order (3,4)
alienation occurring. The social incidence matrix is singular
when you can talk to everyone about everyone's issues (all ones),
or you can't talk to anyone about anyone's issues (all zeros).
When the determinant is 1, it might mean that you can only
if yours is one, you respect privacy of way too many of your
relationships, and there's a good chance you've not got a spiffy
social life. You should probably start introducing these people to each
other unless there is compelling physical danger in doing so, because
you're already in clear and present social danger.

What is important to keep in mind that for n friends there
will be $2^{n^{2}}$ different potential social incidence
matrixes. And that since \textit{alienation} and \textit{affinity}
are related quantities. I'd like to figure out what's the relation
between them in a statistical manner, because they're clearly
not independent quantities. \textbf{Hypothesis}: the alienation
is inversely proportional to the affinity. \textbf{Reasoning}:
If a given person is alienated and alienating, then you're more
willing to bitch about them to everyone you know. Alternately, if you're
very comfortable with some bloke, then you're not going to
go around bitching about them, becaust that would be rude, and
you like this person. So your affinity will be high.

\end{quote}
\end{document}


 Posted by: Jacqueline Russell-Terrier () Posted at: July 11th, 2001 08:04 am (UTC)

You are a very strange and twisted individual.

I like that in a person. :)

 Posted by: antimony () Posted at: July 11th, 2001 09:04 am (UTC)

This is hysterical. And somewhat frightening. And makes me think about modelling more complex interpersonal dynamics with the beam-bending matrix equations, which really ought never be done.

 Posted by: An anonymous kitty () Posted at: July 19th, 2001 07:00 am (UTC) You're such a geek! :^)

I like this. But I think I'd need to feed it into LaTeX--my internal parser isn't up to it this early in the morning. :^)

 Posted by: working on ichi () Posted at: November 9th, 2001 06:58 am (UTC)

That's fuckin awesome.