How do you stay connected with a network when everyone is busy?
I personally do my best to make it to several meetups or networking events each month. I also send out emails or direct messages (DMs) on Twitter to specific individuals. I also set up a bit of time each week to reach back out to reconnect with those I haven't heard from lately.
I send an email or message through social media.
I usually know what sort of commuication an individual prefers. Sometimes an SMS works best.
Catching up online may not be as beneficial as catching up in person, but it sure is powerful, quick and easy.