How to increase my collaboration "network" as a new faculty?
The best way to start I know of is to quietly pick some small problem somebody else is stuck with (requires no social skills whatsoever, just being reasonably open-minded, flexible, and being willing to listen more than to talk), solve it, and communicate the solution to that person. That will open him up immediately and more often than not a bigger collaborative project will ensue. 

In general, keep in mind that most people are willing to talk about what interests them and are quite reluctant to think of what you are interested in, so learn to work on whatever comes your way rather than only on what you specialize in and let it be the other person who chooses the topic and the direction of the collaborative research. Just go with the flow and enjoy the ride. Note that every time you think of or collaborate on something you haven't tried before it is you, who benefits most by gaining new tools and ideas, and the entry cost is usually not very high: most of the time people are stuck on something that can be easily explained in "mathematical common" and requires just a fresh pair of eyes to figure out. 

In summary, rotate the table: create the situation when other people will seek you to ask questions.