A Mathematical Analysis of an Example Delay Tolerant Network using the Theory of Sheaves

NASA’s High-Data Rate Architecture (HiDRA)
project is working towards a general yet practical toolkit and
knowledge base to help usher in the era of new technologies
for space systems communications, such as optical links. The
High-Rate Delay Tolerant Networking (HDTN) implementation
falls under the umbrellas of both the toolkit and the knowledge
base, as its advancements illuminate more general areas of Delay
Tolerant Networking (DTN) that need growth. The goal of
this paper is to explore the usage of particular mathematical
machineries, namely temporal flow networks and sheaves, to
identify fundamental, underlying structures in DTN for space
systems.
Satellites, space assets, ground stations, etc. give rise to a disconnected network, and it is the goal of DTN to glue disparate links
together into a cohesive system, that is, a network. Depending on
a given link, the latencies might be beyond that which the Transmission Control Protocol (TCP) can handle, and contact times
might have one-way light times in excess of minute (sometimes
significantly longer). Some links might be periodic (say, due to
orbital mechanics) or they might not be. This diversity has made
it difficult to probe the underlying structure. An immediate
consequence is that DTNs in practice today are controlled by
globally distributed contact plans (schedules), which are the
input to the contact graph routing (CGR) algorithm. While this
is effective for smaller networks, it will be very difficult to scale
for future networks. Deeper and more rigorous theory is needed
to bring DTN to the next evolutionary step.
To this end, this paper introduces and suggests a mathematical
framework for DTN, and applies it to a space network that is
simulated using an orbital analysis toolkit. The tag-line for the
structure known as sheaves is that they are the mathematically
precise way of gluing local data together into unique, global
data. If we consider routing, we see that networking is a
“sheafy” science. We then discuss a simplified sheaf model,
known as the cellular sheaf. The sheaf-theoretic analysis is presented and discussed, as it is hoped that this and related papers
will help form the primordial ooze of DTN theory. Finally there
is a section of future work suggesting follow-on research.