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We define a new concept, a hyperstructure, which is related to networks and hypernetworks and allows us to represent real problems. We also define the efficiency of this hyperstructure and we apply it to an example. [ABSTRACT FROM AUTHOR]

A novel node-based approach to quantify the vulnerability of a complex network is presented. The proposed measure represents a multiscale evaluation of vulnerability closely related to the edge multiscale vulnerability. In fact, a linear relationship between the edge multiscale vulnerability and the node multiscale vulnerability is stated for p = 1. Upper and lower bounds are established for other values of p. This mathematical framework is subsequently used to obtain some interesting results about the Madrid underground network. [ABSTRACT FROM AUTHOR]

An efficient and computationally advantageous definition of vulnerability of a complex network is introduced, through which one is able to overcome a series of practical difficulties encountered by the measurements used so far to quantify a network's security and stability under the effects of failures, attacks or disfunctions. By means of this approach, we prove a series of theorems that allow to gather information on the ranking of the nodes of a network with respect to their strategic importance in order to preserve the functioning and performance of the network as a whole. [ABSTRACT FROM AUTHOR]

It is important to know how to improve a network when dealing with performance and security. We characterize the optimal design of a network for efficiency or vulnerability giving the extreme networks for those parameters. Moreover, we give an algorithm to improve the vulnerability of a given network when adding a new link. For efficiency, we present two facts that have to be considered in order to improve it when adding a new link, and we calculate bounds for the increment. [ABSTRACT FROM AUTHOR]