Terrorist attacks on transportation networks have traumatized modern societies. With a single blast, it has become possible to paralyze airline traffic, electric power supply, ground transportation or Internet communication. How and at which cost can one restructure the network such that it will become more robust against a malicious attack? We introduce a new measure for robustness and use it to devise a method to mitigate economically and efficiently this risk. We demonstrate its efficiency on the European electricity system and on the Internet as well as on complex networks models. We show that with small changes in the network structure (low cost) the robustness of diverse networks can be improved dramatically whereas their functionality remains unchanged. Our results are useful not only for improving significantly with low cost the robustness of existing infrastructures but also for designing economically robust network systems.percolation | power grid T he vulnerability of modern infrastructures stems from their network structure having very high degree of interconnectedness that makes the system resilient against random attacks but extremely vulnerable to targeted raids (1-17). We developed an efficient mitigation method and discovered that with relatively minor modifications in the topology of a given network and without increasing the overall length of connections, it is possible to mitigate considerably the danger of malicious attacks. Our efficient mitigation method against malicious attacks is based on developing and introducing a unique measure for robustness. We show that the common measure for robustness of networks in terms of the critical fraction of attacks at which the system completely collapses, the percolation threshold, may not be useful in many realistic cases. This measure, for example, ignores situations in which the network suffers a significant damage, but still keeps its integrity. Besides the percolation threshold, there are other robustness measures based, for example, on the shortest path (18)(19)(20) or on the graph spectrum (21). They are, however, less frequently used for being too complex or less intuitive. In contrast, our unique robustness measure, which considers the size of the largest component during all possible malicious attacks, is as simple as possible and only as complex as necessary. Due to the ample range of our definition of robustness, we can assure that our process of reconstructing networks maintains the infrastructure as operative as possible, even before collapsing. ModelModeling Attack on Infrastructures. We begin by demonstrating the efficiency of our unique approach to improve the performance of two of the most fragile, but critical infrastructures, namely, the power supply system in Europe (22) as well as the global Internet at the level of service providers, the so-called point of presence (PoP) (23). The breakdown of any of these networks would constitute a major disaster due to the strong dependency of modern society on electrical power and Internet. ...

Extracting understanding from the growing ``sea'' of biological and socio-economic data is one of the most pressing scientific challenges facing us. Here, we introduce and validate an unsupervised method that is able to accurately extract the hierarchical organization of complex biological, social, and technological networks. We define an ensemble of hierarchically nested random graphs, which we use to validate the method. We then apply our method to real-world networks, including the air-transportation network, an electronic circuit, an email exchange network, and metabolic networks. We find that our method enables us to obtain an accurate multi-scale descriptions of a complex system.Comment: Figures in screen resolution. Version with full resolution figures available at http://amaral.chem-eng.northwestern.edu/Publications/Papers/sales-pardo-2007.pd

We show through a nonlinear Fokker-Planck formalism, and confirm by molecular dynamics simulations, that the overdamped motion of interacting particles at T=0, where T is the temperature of a thermal bath connected to the system, can be directly associated with Tsallis thermostatistics. For sufficiently high values of T, the distribution of particles becomes Gaussian, so that the classical Boltzmann-Gibbs behavior is recovered. For intermediate temperatures of the thermal bath, the system displays a mixed behavior that follows a novel type of thermostatistics, where the entropy is given by a linear combination of Tsallis and Boltzmann-Gibbs entropies.

We investigate the navigation problem in lattices with long-range connections and subject to a cost constraint. Our network is built from a regular two-dimensional (d=2) square lattice to be improved by adding long-range connections (shortcuts) with probability P(ij) approximately r(ij)(-alpha), where r(ij) is the Manhattan distance between sites i and j, and alpha is a variable exponent. We introduce a cost constraint on the total length of the additional links and find optimal transport in the system for alpha=d+1 established here for d=1 and d=2. Remarkably, this condition remains optimal, regardless of the strategy used for navigation, being based on local or global knowledge of the network structure, in sharp contrast with the results obtained for unconstrained navigation using global or local information, where the optimal conditions are alpha=0 and alpha=d, respectively. The validity of our results is supported by data on the U.S. airport network.

We develop a method to generate robust networks against malicious attacks, as well as to substantially improve the robustness of a given network by swapping edges and keeping the degree distribution fixed. The method, based on persistence of the size of the largest cluster during attacks, was applied to several types of networks with broad degree distributions, including a real network—the Internet. We find that our method can improve the robustness significantly. Our results show that robust networks have a novel ‘onion-like’ topology consisting of a core of highly connected nodes hierarchically surrounded by rings of nodes with decreasing degree.