Symmetry in network coding

Abstract

We establish connections between graph theoretic symmetry, symmetries of network codes, and symmetries of rate regions for k-unicast network coding and multi-source network coding. We identify a group we call the network symmetry group as the common thread between these notions of symmetry and characterize it as a subgroup of the automorphism group of a directed cyclic graph appropriately constructed from the underlying network's directed acyclic graph. Such a characterization allows one to obtain the network symmetry group using algorithms for computing automorphism groups of graphs. We discuss connections to generalizations of Chen and Yeung's partition symmetrical entropy functions and how knowledge of the network symmetry group can be utilized to reduce the complexity of computing the LP outer bounds on network coding capacity as well as the complexity of polyhedral projection for computing rate regions.