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by mxuribe 3540 days ago
That is exactly my same use-case: a private/self-hosted social network only for my family; and I happen to use Gnu Social. To be clear, Gnu Social is NOT the only open source platform that accommodates this use-case. Specific to your question, here's a screenshot of some of the settings for restricting access on gnu social: http://imgur.com/a/gXy7y

As far as mobile clients, my family and I use AndStatus [http://andstatus.org] which works awesomely with gnu social, is an extremely lightweight mobile application, gets updated constantly by the dev. team, and also allows you to post to your twitter account.

Good luck!