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\documentclass[10pt]{article} \renewcommand{\rmdefault}{eht} \usepackage{nopageno} \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 talking about your issues with a given person with themselves, 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 talk about your issues with a single person with themselves -- 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}