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 CqCq-stability and the CqCq-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 CqCq-efficient network is not necessarily CqCq-stable for q ≥ 1. We provide sufficient conditions for a CqCq-efficient network to be CqCq-stable.