Please also try checking the box “My computer is permanently connected to the Internet” in Avast Settings → Updates. Otherwise, Avast may be connecting to its servers to find out if a connection is available.

Also, please note that if you have WebShield enabled, pretty much all your HTTP traffic will be routed via AvastSvc.exe. This is because the WebShield works as a proxy (AvastSvc.exe is the proxy process).

Thanks
Vlk