2013 iMac, internet keeps dropping

I'm not sure if this is a Mavericks issue or an iMac issue, as this is my first run with Mavericks. Basically, my internet keeps dropping... sometimes as often as every 10 minutes or so. That aside, my internet no longer seems to connect automatically either (so I have to manually connect regardless whether when starting up, after restarting, or after the connection drops).

I know it isn't an issue with wifi or the wifi card, because my connection is wired. I've never had this problem prior to this (whether with my MacBook Pro or Mini).

Any ideas what's going on? It's bugging the crap out of me because I'll be going about browsing the web or typing up a reply in a forum... go to hit reply and I'd lose my reply because apparently, my connection was dropped without my realizing it. (Like right now, I'm about to hit "Submit" and I decide to double check my connection just in case... and voila, in the time I took to write this post, I've been disconnected!)

 

I'm connected via Ethernet directly to the DSL modem without a router.

Currently, all other computers and devices are connected via wifi, so I can't say for sure whether a wired connection is being dropped on other devices. And those all seem fine.

However, I only just received and began using this iMac a few days ago. Right before it, my Mini was the computer connected directly to the modem and I had never had this problem before. I've also had my MacBook Pro connected this way with no problems either. This only began happening when I started using the iMac (it's also the first and only computer in the house that is on Mavericks).

I think it's mostly happening when the connection is "idling"... I'm not positive about this, but I seem to notice the connection dropping more often when I'm not actively downloading something, refreshing a page, or clicking around. But if it's left hanging while I've taken a little too long reading something or maybe doing something else on the computer that isn't actively using the connection, it will (or at least seem to) drop off more often. It's like the computer noticed the connection isn't being used and decided to just log me off.