A Local Perspective on the Edge Removal Problem

Abstract

The edge removal problem studies the loss in network coding rates that results when a network communication edge is removed from a given network. It is known, for example, that in networks restricted to linear coding schemes and networks restricted to Abelian group codes, removing an edge e * with capacity Re * reduces the achievable rate on each source by no more than Re * . In this work, we seek to uncover larger families of encoding functions for which the edge removal statement holds. We take a local perspective: instead of requiring that all network encoding functions satisfy certain restrictions (e.g., linearity), we limit only the function carried on the removed edge e * . Our central results give sufficient conditions on the function carried by edge e * in the code used to achieve a particular rate vector under which we can demonstrate the achievability of a related rate vector once e * is removed.

Towards an Operational Definition of Group Network Codes

“…The opposite direction representing CWL codes as group codes, was presented (under a slightly different set of definitions) in [15]. The following theorem is given for completeness, and is proven in the Appendix.…”

“…Linear codes are shown to be CWL by choosing the corresponding groups to again be subspaces of the source vector space. Further, the network codes defined by (global) CWL functions are group codes [15], [16]. In this work we address the complementary question asking whether group codes can be represented operationally through CWL functions.…”