I will suppose you have an algebraic taste, and that you want to develop theory rather than analyze data.
A starting point is the observation that relationships in a social group can be modeled by a semigroup. Among the leaders of this area, developed around the 70's, you can find Harrison White. He completed a PhD in theoretical physics at MIT, then a PhD in sociology at Princeton. Currently, he is at Columbia. A serious trouble of this approach (from my point of view) is that relationships change over time, so we do not have a stable algebraic structure. There are ways around it... If you want to follow this lead, then I recommend you the book Algebraic Models for Social Networks by Philippa Pattison.
Another observation is that social sciences are full of relationships we don't measure with numbers. So, a categorical-functorial approach has been developed. In that sense Lorrain and White wrote an enjoyable article:
http://www.tandfonline.com/doi/pdf/10.1080/0022250X.1971.9989788
I would like to know how to use some category theory in this context...
I must say many sociologists dislike this approach a lot, because they consider it reductionist. I am afraid research in this area may fall in no man's land i.e neither mathematicians nor sociologists read it. However, there are some groups trying to develop this field. In particular, UC Irvine has the Institute for Mathematical Behavioral Sciences: http://www.imbs.uci.edu/. I also recomend to look up for Santa fe institute.
On other hand, a big problem for social sciences is measurement. Suppose, you develop a social theory (for example [1]). To validate the model you need to test it with reality. This is a big issue in social sciences, because you cannot take people inside a box, isolated other variables, and measure a specific behavior. Plus, real life events have many variables to account for. However, these days Google, Facebook, NSA, etc, are collecting a lot of social generated data; and there is not good theory for this sort of data. The void is being filled by computer scientists; for example Carnegie Mellon has a PhD program at its Center for Computational Analysis of Social and Organizational Systems. I attended their main conference some time ago...It impressed me their focus on extract, analyze, and represent relational data rather than understand it (in the sense of constructing theory). That perspective can be very useful, but I dislike it. Hopefully, social generated data can be used one day for constructing theories. (Noam Chomsky argued for developing theory here:
http://www.theatlantic.com/technology/archive/2012/11/noam-chomsky-on-where-artificial-intelligence-went-wrong/261637/).
[1] "A New Model of Wage Determination and Wage Inequality." Rationality and Society, 2009 by Guillermina Jasso
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