There are allot of possibilities as to why limelight networks, llc is connecting to you. It could be software on your machine. I did a search on them If you have Macromedia on your machine there is a good chance that could be the cause. Along with a host of others. You might find the link below interesting.
http://www.bedagi.com/news/3494
I Have LimeWire Pro on my machine below is the log I took a few minutes after I shut it down.
C:\Documents and Settings\Rick>NetStat -a -n
Active Connections
Proto Local Address Foreign Address State
TCP 0.0.0.0:135 0.0.0.0:0 LISTENING
TCP 0.0.0.0:445 0.0.0.0:0 LISTENING
TCP 0.0.0.0:3260 0.0.0.0:0 LISTENING
TCP 0.0.0.0:3261 0.0.0.0:0 LISTENING
TCP 127.0.0.1:1039 0.0.0.0:0 LISTENING
TCP 127.0.0.1:2990 127.0.0.1:12080 CLOSE_WAIT
TCP 127.0.0.1:3836 127.0.0.1:12080 CLOSE_WAIT
TCP 127.0.0.1:4047 127.0.0.1:12080 CLOSE_WAIT
TCP 127.0.0.1:12025 0.0.0.0:0 LISTENING
TCP 127.0.0.1:12080 0.0.0.0:0 LISTENING
TCP 127.0.0.1:12110 0.0.0.0:0 LISTENING
TCP 127.0.0.1:12119 0.0.0.0:0 LISTENING
TCP 127.0.0.1:12143 0.0.0.0:0 LISTENING
TCP 192.168.1.100:139 0.0.0.0:0 LISTENING
TCP 192.168.1.100:2927 192.168.1.102:445 Microsoft ESTABLISHED
UDP 0.0.0.0:445 *:*
UDP 0.0.0.0:500 *:*
UDP 0.0.0.0:1025 *:*
UDP 0.0.0.0:1026 *:*
UDP 0.0.0.0:1040 *:*
UDP 0.0.0.0:1063 *:*
UDP 0.0.0.0:1845 *:*
UDP 0.0.0.0:4500 *:*
UDP 127.0.0.1:123 *:*
UDP 127.0.0.1:1029 *:*
UDP 127.0.0.1:1033 *:*
UDP 127.0.0.1:1900 *:*
UDP 127.0.0.1:2979 *:*
UDP 127.0.0.1:3809 *:*
UDP 127.0.0.1:4030 *:*
UDP 192.168.1.100:123 *:*
UDP 192.168.1.100:137 *:*
UDP 192.168.1.100:138 *:*
UDP 192.168.1.100:1900 *:*