This I have so far made clear to myself. Your cable modem does not have any wireless capabilities. Therefore there is no way a person can have access your network without a possible backdoor Trojan being installed on your computer system.
You have formatted your hard drive several times and have re-installed your operating system since. Usually if a Trojan is accidentally installed on a computer system, a deep format usually removes all traces of the Trojan.
There are boot sector Trojans, however, those boot sector Trojans are wiped during the disk format procedure.
If you feel for any reason, that a Trojan is hiding on a portion of your hard drive, you can destroy any possible remnants of such by using a DOS boot disk that allows you to boot up your computer independent of your hard drive. While using a DOS boot disc, you can delete the disc partitions. I would delete all partitions including any system reserved ones.
When all the partitions have been deleted, your hard drive is back to factory default status. Installing a bootable operating system such as Windows 2000, Windows XP, Windows Vista, Windows 7 or Windows 8, automatically formats the disk, adds the boot sector information, which includes the files and directory locations that are located across the disc platters.
A boot sector Virus or Trojan infection is usually wiped clean during this process. Now, there are rare instances where a Virus contains firmware. Firmware is code that can be flashed into a programmable chip. In fact, all computer hardware works via firmware. There are viruses that can be installed that flash a hardware's firmware related sector of the chip. This often happens when people try to install drivers for their hardware that came from a website that hosts P2P file sharing.
The problem with older hardware, is that current drivers for such, are no longer available or the company went out of business and customer support is no longer available. People result to using search engines--such as Google or Bing--to locate what seems to be newer drivers that support todays operating systems. Often than not, those people download drivers from CNET or Peer-to-peer also known as P2P file hosting sites. You have no clue if the drivers are legit ones written by your hardware vendor's code writers. Those drivers are in all likelihood, written by some script kiddie, who took the original code and renamed it with a higher version number and added some bad code that installs a backdoor Trojan or flashes your hardware's firmware.
Tryingmybest, I hope I did not go to crazy here with my analysis and explanation of how things work. I have spent 35 years in the electronics repair field and I am currently attending college for a computer science degree to learn how to write code to produce software programs for computer usage. I spend many hours and have spent many years learning--what I can tech someone else in less then an hour.