Does the size of our social network correlate with brain growth?

It seems so, although more research still has to be done using human subjects.

A research group from the University of Oxford used macaques to study this. They placed them in groups of different sizes during one year.
After this period, parts of the brains (important for recognition of faces, as well as facial and emotional expressions) had increased in the monkeys that had been living in bigger groups. These functions seem to be needed harder to be able to be successful in a bigger social context.

Read more at:

http://www.sciencemag.org/content/334/6056/697.abstract