Abstract
Information does not generally behave like a conservative fluid flow in communication networks with multiple sources and sinks. However, it is often conceptually and practically useful to be able to associate separate data streams with each source–sink pair, with only routing and no coding performed at the network nodes. This raises the question of whether there is a nontrivial class of network topologies for which achievability is always equivalent to ‘routability’, for any combination of source signals and positive channel capacities. This chapter considers possibly cyclic, directed, errorless networks with n source–sink pairs and mutually independent source signals. The concept of downward dominance is introduced and it is shown that, if the network topology is downward dominated, then the achievability of a given combination of source signals and channel capacities implies the existence of a feasible multicommodity flow.
A short, preliminary version without proofs was presented in the conference paper [14].
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Notes
- 1.
Such digraphs are sometimes called simple.
- 2.
If a source or sink were actually connected to multiple nodes in the network, it would be represented in the digraph by an auxiliary vertex connected by an arc (of infinite capacity) with a multiply-connected vertex.
- 3.
For instance, when a single network node is represented as two ‘virtual’ vertices connected by an arc of unbounded capacity.
- 4.
In the linear, time-invariant context of [17], this is equivalent to the corresponding transfer functions being well-defined and proper.
- 5.
In other words, if S solves (Σ,X,c), then there exist variable bit-rate codes for each arc that yield errorless, block-by-block reconstruction of the source-signals at their sinks, with expected bit-rates at worst negligibly larger than arc-capacities. Conversely, if there exists a distributed entropy coding scheme that achieves perfect reconstruction of source-signals at their sinks in blocks of length m, and with expected bit-rates no larger than the arc-capacities, then this yields a solution S as defined above. However, these operational interpretations will not be used in this article.
- 6.
Ignoring differences in the definition of achievability.
- 7.
Here, arcs are permitted to have r α =0.
- 8.
If an arc is not on any i-path, then its arc signal may be taken to be 0.
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Acknowledgements
The author is indebted to the reviewers for numerous helpful suggestions. He also acknowledges discussions on the decentralized control version of this problem with Prof. Rob Evans at the University of Melbourne.
This work was partially supported by Australian Research Council grant DP110102401.
This research was supported by LCCC—Linnaeus Grant VR 2007-8646, Swedish Research Council.
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Nair, G.N. (2014). Structural Routability of n-Pairs Information Networks. In: Como, G., Bernhardsson, B., Rantzer, A. (eds) Information and Control in Networks. Lecture Notes in Control and Information Sciences, vol 450. Springer, Cham. https://doi.org/10.1007/978-3-319-02150-8_7
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