The effect of removing a network communication edge: Group network codes

The edge removal problem quantifies the loss in rate when removing an edge from a given network. In this work, we study the edge removal problem on network coding instances that are solvable using group network codes. We show that removing any edge of capacity Re from a given network reduces the rate vector achievable by abelian group network codes by at most an additive Re. Our work extends previous results of similar nature on linear network codes which are a special case of group network codes. The extent to which the achievable rate is affected by removing an Re capacity edge, in the presence of general network coding functions, is yet to be quantified.

[1]  Tracey Ho,et al.  On equivalence between network topologies , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[2]  T. Chan,et al.  Capacity regions for linear and abelian network codes , 2007, 2007 Information Theory and Applications Workshop.

[3]  Michael Langberg,et al.  Network coding: Is zero error always possible? , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[4]  Raymond W. Yeung,et al.  On a relation between information inequalities and group theory , 2002, IEEE Trans. Inf. Theory.

[5]  Terence Chan,et al.  Group characterizable entropy functions , 2007, 2007 IEEE International Symposium on Information Theory.

[6]  Terence Chan.  On the optimality of group network codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[7]  Raymond W. Yeung,et al.  Information Theory and Network Coding , 2008 .

[8]  Michael Langberg,et al.  Outer bounds and a functional study of the edge removal problem , 2013, 2013 IEEE Information Theory Workshop (ITW).

[9]  Terence Chan,et al.  Dualities between Entropy Functions and Network Codes , 2008 .

[10]  Michael Langberg,et al.  On a capacity equivalence between network and index coding and the edge removal problem , 2013, 2013 IEEE International Symposium on Information Theory.

[11]  Shuo-Yen Robert Li,et al.  Linear network coding , 2003, IEEE Trans. Inf. Theory.

[12]  Alex J. Grant,et al.  Network Coding Capacity Regions via Entropy Functions , 2012, IEEE Transactions on Information Theory.

[13]  Muriel Médard,et al.  An Edge Reduction Lemma for linear network coding and an application to two-unicast networks , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[14]  Tracey Ho,et al.  On the impact of a single edge on the network coding capacity , 2011, 2011 Information Theory and Applications Workshop.

[15]  Zhen Zhang,et al.  The Capacity Region for Multi-source Multi-sink Network Coding , 2007, 2007 IEEE International Symposium on Information Theory.