The Egg Network, or Eggnet, is a global communications network used by the Eggman Empire to keep in touch with its various territories across Mobius. It was while performing a routine maintenance check on the Sub-Bosses via the Eggnet that Snively Robotnik came into contact with Regina Ferrum, the Iron Queen, who he became romantically attached to, eventually culminating in them usurping control of the Eggman Empire from Dr. Eggman. (StH: #192, #194, #200)
Later on, Mighty the Armadillo and Ray the Flying Squirrel of the Chaotix accessed the Eggnet for information on the whereabouts of Mighty's sister. When the duo went missing in action, their teammates Vector, Espio, and Charmy also used the Eggnet to help in finding them. (SU: #47, #48)
After the events of Worlds Collide, it was via the Eggnet that Dr. Eggman discovered the damage he had caused to Mobius's reality by disrupting Super Sonic's attempt to reverse the Super Genesis Wave's effects. (StH: #253).