On secure network coding for two unicast sessions: Studying butterflies

This paper considers a class of wireline networks, derived from the well-known butterfly network, over which two independent unicast sessions take place simultaneously. The main objectives are to understand when network coding type of operations are beneficial with and without security considerations and to derive the gains that cooperation among sources and sinks can bring. Towards these goals, the capacity region of the butterfly network with arbitrary edge capacities is first derived. It is then shown that no rate can be guaranteed over this network under security considerations, when an eavesdropper wiretaps any one link. Four variations of the butterfly network, such as the case of co-located sources, are analyzed and their secure and non-secure capacity regions are characterized.