Connectivity, stability and efficiency in a network as an information flow

We consider the problem of forming a network as an information flow under the requirement that the whole network be connected and remain connected after the destruction of some nodes. We introduce the C^q-stability and the C^q-efficiency of a network which, roughly speaking, require connectivity in addition to stability and efficiency even after any q nodes are destroyed. We mainly examine the relation between efficiency and stability. With the connectivity requirement, the efficient network is always stable, but the C^q-efficient network is not necessarily C^q-stable for q ≥ 1. We provide sufficient conditions for a C^q-efficient network to be C^q-stable.