hey and welcome to the forum. what version of avast you using? free pro suite?
you could try a installation from the scratch and see if that could help.
Please note that a torrent program such as LimeWire will open
100's/1000's/10,000's of internet connections out of your computer, as that is what it is designed to do. Avast Web Shield would try to cover all these connections, Network Shield, as well.
What is notable is that this behavior is from the get-go, it begins from the moment your computer is started from a cold start. Uninstalling a torrent program will help to fix that, and speed up your computer. Much less work for your computer to have to do.
Do
mikaelrask's advice, then do the second. What version of firewall are you running?
Does sound like you have an infection.
Post logs from this link here:
http://forum.avast.com/index.php?topic=53253.0 Attach all logs, run Malwarebytes, OTL, and aswMBR.exe only, and
stop all cleaning efforts for now. You can damage your system more than what it already is.