Assuming logging in with Facebook/Google etc. gives the social media company access to and ability to record all the interactions of the user on your website, how does this affect the behaviour of the user on your website?
I have surprisingly noticed the following behavior with my social logins.
You may or may not use this to your advantage if you have multiple logins on your website which users need to select from.
If I happen to be on a Fun website which is about connecting with people/ friends/ family/ etc, I generally would sign in with Facebook.
If it's something related to professional work, career, courses, etc, I would sign in with LinkedIn or G+.
If it is a sketchy website which I do not trust, and is only for a one time visit, I would sign in with my Email (Secondary).
You may calculate your user's interest or loyalty to your website with these in mind, according to the category your website falls in.
So, to answer your question, the behavior of the user is altered, not as per the social network he chooses but the website he chooses the particular social network for.
I am inclined to consider this as a valid question in a grander concept of user experience. Naturally users are affected by which site log ins are accepted / offered. What follows is strictly my personal opinion, no data to back this.
There was a time when Facebook ubiquitous everywhere. Since then there is a strong perception of Facebook with leisure and social activities. May it be candy crush or Farmville, a Facebook authentication has become analogous to something which probably has to do with leisure and something where your friends / social interactions matter.
Google came very late in the game, but since then, they have carved up a nice niche of sites which offer Google log in. Initially it appeared that Google is playing catch up game with Facebook in the hope to bag more and more sites with their authentication. However things took a different turn when sites like Asana, started offering only Google sign in. Asana is a productivity app, with an air of enterprise task management. Google's enterprise suite capital worked in their favor and somehow Asana and Google log in was a snug fit. I don't recall, but I have seen few more sites using only Google authentication, especially those which are interested in just identity and not the social angle.
So you have Facebook sign in with inclination towards social and leisure, and you have Google with a tinge of enterprise / business orientation. There are likes of Twitter, but I don't really know where I would place them.
So to answer your question, it should matter to you which site log ins you are offering and for what reason. People may infer the tone of your site based on that. It might not be a make or break decision for the user, but she would probably consider it.