BuddyPress.org

It really depends what you mean by “customization”. If you mean that each group should be able to load its own background/header/color scheme, then it’s definitely possible. I don’t know of a plugin that does it, but it wouldn’t be all that hard to write something that allowed group admins to switch between a number of admin-selected color schemes, or to upload images in addition to the avatar.

Is there some other meaning of “custom network” that I’m missing out on here? Something that “private/hidden” doesn’t already cover?