The fig on page 11 of the rules implies that you can only remap a tile of a network in which you have an avatar as there are three pendant tiles but only the two within the network the avatar is in are shown to be remappable. Is this correct?
No - you do not have to have an avatar on the network. The illustration would be better if the upper left server tile was also dark blue.
One of the fun things in Cyberspace is remapping your opponent into a corner and then getting them stuck there by adding another tile to lock them in place.