The power of weak ties
Sociologist Mark Granovetter wrote a classic publication in 1974 called Getting a Job. He has studied how people had actually found their current job and found out that 56% of them had found it by networking, 18,8% through formal channels (job openings, recruiters), and 20% by applying directly. Furthermore, he found that of the people who had found their job through networking (which the majority of the people did), 16,7% found it through a close relationship (a "strong tie"), someone whom they met a least once a week, 55,6% found it through a relationship they met only occasionnally, more than once a year but less than twice a week (a "weak tie"), and 28% through a relationship they met with only rarely, once a year or less (an insignificant tie). The surprising conclusion is: people usually don't find their new jobs neither through formal channels, nor through close friends and relationships. Instead they find them through weak ties, people they meet only occasionally, people in the periphery of their personal network. The reason is your friends and close relationships move in the same circles as you and see and know roughly the same as you. What they know, you are likely to know too already. But the weak ties people introduce you into new worlds. They have links to new networks and see what you don't see. Therefore, they may be the people who may catapult you into new environments.
An example of using the power of weak ties
Once you realize the importance of weak ties, it is relatively easy to start using this knowledge. Here is one approach you might try. First, write a simple and brief e-mail text in which you a) greet the person, b) say that you are looking for a new job and explain what kind of job or working environment you search, c) ask the person if he or she knows someone you might talk to about this. To give you just an idea of what such an email might look like, here is an example: