The only way to answer this professionally is a full interview and analysis of the facts, not your hypothetical. Less than that is unethical on our part and would screw you. But such an interview cannot ethically be carried out in a public forum. That means you need to get a lawyer. Many, me included, will do a brief discussion by phone for free.
Licensed in Maryland with offices in Maryland and Oregon. Information here is general, does not create a lawyer-client relationship, and is not a substitute for consulting with an experienced attorney on the specifics of your situation.
Based upon your hypothetical, I'd say the risk is huge that you're going to get sued. If a big company with lots of resources is trying to build its brand in a social media space, it is going to do what it can to protect the distinctiveness of that brand. From your description of the matter, it looks like your case for noninfringement wouldn't be very strong. But even if you had a 50/50 case for noninfringement, the big company is still likely to take some action. And do you really want to be slogging through litigation when you are just trying to start your company?
Of course there's more to it! Plus, we don't have an attorney-client relationship. This brief comment is for information only, and must not be relied upon as legal advice.
HANG TEN is a famous brand with 15 pending or issued registrations, many incontestable. It is one of the world's most famous beachware brands. So, I doubt that it is a social gaming network, as the owner of that would likely be sued for trademark infringement and trademark dilution. So, your hypothetical is nonsensical.
If your question is as it seems: "If I adopt a famous social gaming network servicemark of a bigger company as the name of my new app, would they be likely to litigate, settle or leave me alone?" Big businesses have lawyers that would be more than happy to sue you for doing that and who would be worried for their career if they failed to sue you.
No, I won't refer you to recent trademark litigation precedents because it would be a disservice to you. You can't do this yourself. Be smart, if you have to ask such a question you have no business serving as your own lawyer. Don't try. See a trademark lawyer and quit trying to do it yourself before you end up getting yourself in a very expensive legal mess. It is much cheaper to hire that trademark attorney now to keep you out of trouble rather than later to get you out of trouble, which might easily cost you ten times as much.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Other attorneys have answered the questions fully.
My contribution: The purpose of this forum is not to educate you by providing trademark litigation precedents. Such information requires experience and research requiring attorney time. Such research will also be based on your situation which, if you consult an attorney in person, you will find out needs more facts than you describe here. Your expectation to ask for such information is unreasonable.
You must contact a local attorney or obtain telephonic consultation from an attorney to discuss specifics of your situation and get full advise. The information given here is general in nature and does not establish attorney-client relationship.