Sounds like wake on lan. WOL.
Try going into your bios and see if there is an option for it.
Or, check your network interface card's manual to see if there is this feature.
Also, to just test the issue further, remove your ethernet cable from the back of the machine, leave it off and see if it does the random start.
If there is no ethernet cable connected, you don't have a lan connection and therefore WOL should not start the machine.
Remember, there are all sorts of nasty people out there with bad intention and some may go as far as to try to turn your machine using WOL to try and take over your connection/machine.
A firewall may defeat this feature, I am not certain though.
Good luck.
(If this turns out to not be related, I apologize, but it is certainly a possibility.)
Gentle