So, I've read that logging out is done like this:
In my point of view, they simply remove the local access token and any cached data. By looking at the comments of the logout method from the sdk file:
/*! @abstract Logs the user out @discussion This calls [FBSDKAccessToken setCurrentAccessToken:nil] and [FBSDKProfile setCurrentProfile:nil]. */ - (void)logOut;
We as a developer-user have to rely on how they provide the api, and from the this method, it looks like its a local job (client-side), no web service involved. So there's no need to show activity indicator or whatsoever.