In this paper we develop a preliminary model for social networks , and a measure of the level of polarization in these social networks, based on Esteban and Ray's classic measure of polarization for economic situations. Our model includes information about each agent's quantitative strength of belief in a proposition of interest and a representation of the strength of each agent's influence on every other agent. We consider how the model changes over time as agents interact and communicate, and include several different options for belief update, including rational belief update and update taking into account irrational responses such as confirmation bias and the backfire effect. Under various scenarios, we consider the evolution of polarization over time, and the implications of these results for real world social networks.