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 Cq-stability and the Cq-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 Cq-efficient network is not necessarily Cq-stable for q≥1. We provide sufficient conditions for a Cqefficient network to be Cq-stable. © 2007 Elsevier B.V. All rights reserved.