Imagine if you join a new hobby group. Upon arrival you were asked to text a single message all your contacts inviting them to come. Would you do it? How many would come?
I'm guessing you answered no and not many.
When someone joins your community, don't prompt them to invite everyone they know to join.
In fact, don't use automated prompts at all. It fails far more frequently than it succeeds. It may work for Twitter, but it won't work for you.
Automated prompts are too forced, too impersonal, and lack a reason for a contact to join.
We created an invite your friends option on CommunityGeek last year. No-one used it.
Yet, referrals are the best way to grow a social group. Your job isn't to force people to invite their friends. Your job is to use basic psychology to motivate them to invite their friends.
People will invite/refer friends for one of two reasons. Either it helps their friend or it makes the existing member look good.
To achieve the former, you need to have member-curated resources that you can turn into definitive guides to attract newcomers. Newcomer and advanced guides to the topic work especially well here. Guides or areas of the site dedicated to specific, common, problems are also good.
To achieve the latter, you need to give every member a sense of autonomy and ownership over their area of the community. For example, anyone interviewed by a community shares that interview on Facebook/Twitter. Anyone that shares a personal experience or any niche expertise they have is also more likely to share it with others.
The greater the level of autonomy (freedom to act as they wish) and ownership (responsibility for areas of the community) you give every member - the more you will find they bring other members in.
No need to game or rig this system, just use the psychology.