Range of human radio broadcast.

Not the square, but the blue dot. 

I have heard that our radio broadcasts are too weak to extend even that far.

This is about three times the diameter of "Known Space", which is the area explored by humans in a fictional series of great science fiction novels by Larry Niven and others.  https://en.wikipedia.org/wiki/Known_Space  However, at least five different advanced species exist in "Known Space."

My point is that you could have a vast empire just in the area of the blue dot.

BTW, our galaxy has so much mass that there are a couple of smaller galaxies that orbit it in the same way planets orbit the sun.