According to RFC1925, the second fundamental truth of networking is: No matter how hard you push and no matter what the priority, you can’t increase the speed of light.
However early in the world of network engineering this problem was first observed (see, for instance, Tanenbaum’s “station wagon example” in Computer Networks), human impatience is forever trying to overcome the limitations of the physical world, and push more data down the pipe than mother nature intended (or Shannon’s theory allows).
One attempt at solving this problem is the description of an infinitely fat pipe (helpfully called an “infan(t)”) described in RFC5984. While packets would still need to be clocked onto such a network, incurring serialization delay, the ability to clock an infinite number of packets onto the network at the same moment in time would represent a massive gain in a network’s ability, potentially reaching speeds faster than the speed of light. The authors of RFC5984 describe several attempts to build such a network, including black fiber, on which the lack of light implies data transmission. This is problematic, however, because a lack of information can be interpreted differently depending on the context. A pregnant pause has far different meaning than a shocked pause, for instance, or just a plain pause.
The team experimenting with faster than light communication also tried locking netcats up in boxes, but this seemed to work and not work at the same time. Finally, the researchers settled on ESP based forwarding, in which two people with a telepathic link transmit data over long distances. They compute the delay of such communication at around 350ms, regardless of the distance involved. This is clearly a potential faster than speed-of-light communication medium.
Another plausible option for building infinitely fat pipes is to broadcast everything. If you could reach an entire region in some way at once, it might be possible to build a full mesh of hosts, each transmitting to every other host in the region at the same time, ultimately constituting an infinitely fat pipe. Such a system is described in RFC6217, which describes the transmission of broadcast packets across entire regions using air as a medium. This kind of work is a logical extension of the stretched Ethernet segments often used between widely separated data centers and campuses, only using a more easily accessed medium (the air). The authors of this RFC note the many efficiencies gained from using broadcast only transmission modes, such as not needing destination addresses, the TCP three-way handshake process, and acknowledgements (which reportedly consume an inordinate amount of bandwidth).
Foreseeing the time when faster than speed-of-light networking would be possible, R. Hinden wrote a document detailing some of the design considerations for such networks which was published as RFC6921. This document is primarily concerned with the ability of the TCP three-way handshake to support an environment where the network’s speed of transmission is so much faster than the speed at which packets are processed or clocked onto the network that an acknowledgement is received before the original packet is transmitted. R. Hinden suggests that it might be possible to use packet drops in normal networks to emulate this behavior, and find some way to solve it in case faster than speed-of-light networks become generally available—such as the ESP network described in RFC5984.
More recent, and realistic, work in faster than speed-of-light networking has been undertaken by the proposed Quantum Networking Research Group in the IRTF. You can read the proposed architecture for a quantum Internet here.