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Abstract: In this paper we propose a mechanism for the formation of paths of minimal length between two points (trails) by a collection of individuals undergoing reinforced random walks. This is the case, for instance, of ant colonies in search for food and the development of ant trails connecting nest and food source. Our mechanism involves two main ingredients: (1) the reinforcement due to the gradients in the concentration of some substance (pheromones in the case of ants) and (2) the persistence understood as the tendency to preferably follow straight directions in the absence of any external effect. Our study involves the formulation and analysis of suitable Markov chains for the motion in simple labyrinths, that will be understood as graphs, and numerical computations in more complex graphs reproducing experiments performed in the past with ants. [Copyright &y& Elsevier]

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]

We define a new general framework, the family of (psi,p,q)-vulnerabilities, as a tool that produces many new vulnerability functions that measure the capacity of a network to maintain its functional performance under random damages or malicious attacks. This new framework comprises most of the vulnerability definitions appearing in the literature and allows to calculate some relationships between the different (psi,p,q)-vulnerabilities in terms of their function psi or their parameters p, q that improve several known results for the vulnerability functions. Some graphics of simulations are provided in order to show the sharpness of these relationships.