Completing circuits with transistors and arduino
How many connections are required?  Do you need a full cross-point matrix or is there only a few connections needed?  I'm defining "few" as less than 32 relays.

If there are only a few connections needed, I'd probably use relays for this purpose.  They are inexpensive and robust.  You can control pretty much any number of relays from you Arduino by using a simple SPI connection to a bunch of TPIC6595 serial shift registers.  Each shift register will directly drive 8 relays and you simply cascade the shift registers for as many relays as you need.  The SPI interface requires 3 wires from your Arduino.