Ok. I think I may have figured this out (at least for my setup).
I used to get the orange download arrow lighting up every 2-3 seconds both at home and at my office and I couldn't figure out why. I would not get this behavior if connected to the mobile data network. Turns out that the router was at issue. I used the same brand and model of router at home and at the office (Asus RT-N16).
Apparently, the default Asus firmware has something called "STP"
(Spanning Tree Protocol) enabled. There is no way to disable this in the Asus firmware. I used WireShark to look at the packets on a router with absolutely nothing connected and this STP packet kept coming in repeatedly. As far as I can tell, there is no reason to have this enabled on a home network. Even on a corporate network, it doesn't seem like it is needed (but I'm not an expert).
I don't know if Linksys/Cisco equipment has this enabled by default but you can take a look in the firmware to see if there's an Enable/Disable button. If you don't see anything, try running WireShark and look for Spanning-tree-(for-bridges) under the destination. If you see that pop up constantly, it means it's enabled. It also helps to do this on a router that's only connected to your computer so you can more easily scan the traffic.
To combat this (and see if it makes any difference in battery life), I've installed DD-WRT on the Asus router. DD-WRT has an option to disable STP and it's disabled by default. So far, so good. The constant blip of the orange arrow is gone and I don't see any STP lines in WireShark.
Hope this helps someone out there!