In Russian, the term "sv'ázi" ("связи") is used to describe something like business connections but it is not that exactly. "Sv'ázi" usually does not refer to friends made through business, but it normally refers to someone you know who could provide you with a service in an unofficial capacity. Russian phrase sounds like "Someone has sv'ázi somewhere" and means that he or she has friends or relatives working in mentioned organization and they could help to achieve an extra or better service from this firm or state organization like getting something without waiting a common queue or simply more quickly, cheaper, getting a job and so on. So knowing them is not a consequence of your business, but because you know them, you can get something done.
Is there an English word or idiom that captures this idea of having friends or family in an organization to help turning interaction with this organization more easy and effective than in official common way?