A few solutions.
1) As mentioned in comments already, the base of the network is cryptography. Every recipient is identified by a public key. Assuming no two keys the same, just one person is identified uniquely. This also prevents man-in-the-middle attacks. Addressing a public key means you know the public key. If you address a man-in-the-middle, how would he know to whom you want to write?
That's the advantage to have cryptography a key component of the network architecture instead of just layering it on top. Another advantage is that there is no centralized organization governing the addresses. If we use variable length RSA, then you could extend the space to whatever bit-count you want. As nobody on the route needs to decode your RSA address, just the end points, you could very well go with 16384bit if you need to.
2) Routing is done by the client. It's easy to intercept your messages if they're managed by unknown hops. it's also a high load for the hops to identify the route for every packet. It's also extremely energy inefficient as this is done again and again. It might look like impractical at first, but crypto currencies let you download Gigs of transactions, which is basically a routing tree for money transfers, you could imagin it very alike for Internet 2.
It would work like a gift that is recursively wrapped. let's say we have nodes A,B,C...Z to your target Z and every node has a public key: a,b.c....z to transfer your message, you'd reversely wrap your text and attach the key to the target. (Assume that * is the encoding operator and + is the attach operator)
now you send your packet to the first node 'A'. it decodes your message with its private key a'. it reads the attachment, which is 'b' the public key aka address of the next node. 'B' also decodes the new packet with its private key b', reads the attached address etc. until it arrives at 'Z'.
You see, nobody can decode the whole route, nobody knows the real target as the package could even be send around the world back to the origin. Nobody can manipulate the final message without knowing the private key of the recipient. And the limit in size automatically limits the age a packet can reach. once it is 0 in size, there is nothing to route.
There is just one real validity check at the end. if a node decodes an attachment that points to itself, it's was the target. If there is a transaction error, the decoded packet will have an address of the next hop that is totally nonsense, thus unknown to the current node.
3) It needs to be a totally unified ad hoc network. This guarantees that nobody owns it, there is no real hierarchy that could be exploited and there is no weak link in the chain. The security of the data would rely on the RSA security and the security of the device of both ends, even if you'd transfer the message on a postcard, it would be safe.
Sitting next to a person, sending data, you wouldn't need to send it to a cloud, across countries and security gates, it would be a direct connection and even then totally encrypted. It could be done via blue tooth, wifi direct, fire wire, usb...you could even save data on a memory stick and pass the stick to the recipient who'd decode the packets.