  A family of equivalence tools for bounding network capacities is introduced.
Part I treats networks of point-to-point channels. The main result is roughly
as follows. Given a network of noisy, independent, memoryless point-to-point
channels, a collection of communication demands can be met on the given network
if and only if it can be met on another network where each noisy channel is
replaced by a noiseless bit pipe with throughput equal to the noisy channel
capacity. This result was known previously for the case of a single-source
multicast demand. The result given here treats general demands -- including,
for example, multiple unicast demands -- and applies even when the achievable
rate region for the corresponding demands is unknown in the noiseless network.
In part II, definitions of upper and lower bounding channel models for general
channels are introduced. By these definitions, a collection of communication
demands can be met on a network of independent channels if it can be met on a
network where each channel is replaced by its lower bounding model andonly if
it can be met on a network where each channel is replaced by its upper bounding
model. This work derives general conditions under which a network of noiseless
bit pipes is an upper or lower bounding model for a multiterminal channel.
Example upper and lower bounding models for broadcast, multiple access, and
interference channels are given. It is then shown that bounding the difference
between the upper and lower bounding models for a given channel yields bounds
on the accuracy of network capacity bounds derived using those models. By
bounding the capacity of a network of independent noisy channels by the network
coding capacity of a network of noiseless bit pipes, this approach represents
one step towards the goal of building computational tools for bounding network
capacities.
