I am working on a mobile app that will have a friends and follower feature. Should a user's profile page display how many friends they have and how many people are following them? What advantages and disadvantages are there to displaying the count? Should a user be more concerned with who is in their network and less about how many people are in their network?
Stats and numbers are what distinguish novice users from experienced ones. It's a reward for having been active on the platform and using it more than another person might have.
Specific to your application, you've mentioned that a user can have Friends as well as Followers. Looking at a few examples of such an implementation, Facebook introduced Subscribers in 2011 in addition to Friends.
- Quick indicator of the person's status on the platform. Older, or more influential users are likely to have higher counts.
- A decision maker for users deciding whether they should follow a person or not. The count is correlated to the quality of posts.
- Useful as a distinguisher in the case of duplicate profiles of influential users. User is more likely to follow the more active profile.
- The user might not want a certain stat to be public. Facebook allowed the ability to hide friend list and count in late 2009.
- If your model depends upon paying to see certain privileges. LinkedIn for example shows connections and parts of the profile only if you have a connection.
Every social network out there displays, at least to the user, the number of friends/followers they have. People like to feel accomplished. People see a higher number of followers/friends as that accomplishment in a social network, especially content creators. People who go out of their way to create content for your site gauge their "worth" as a participant of the site based on how many likes/reblogs/shares/notes/etc they get based on that content, and in turn how many people follow/friend them as a result of creating that content.
So yes. You do need a friend/follower count. It retains users. Without it no one feels the need to come back.
As wtfsven and abhinavc said the numbers help distinguish users but it does so by rewarding a certain type of behaviour. You asked whether a social app can survive without such numbers and I would say yes, it can. But would you really want to do without them? I don't know what your social network revolves around but there is plenty of options to use a quantifier to reward a type of behaviour that you want to endorse. StackExchange rewards contributions by tying reputation gained from answers to power within the community. Number of followers/friends is just one attribute that can be used but in my opinion there are many other options that are more appealing.
TL; DR. Use points/achievements/badges to reward the user behaviour that helps evolve your network.