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.

Many natural and social systems display global organization and coordination without centralized control. The origin of this global coordination is a topic of great current interest. Here we investigate a density-classification task as a model system for coordination and information processing in decentralized systems. We show that sophisticated strategies, selected under idealized conditions, are not robust to environmental changes. We also demonstrate that a simple heuristic is able to successfully complete the classification task under a broad range of environmental conditions. Our findings hint at the possibility that complex networks and ecologically efficient rules coevolve over time.

We investigate flux front penetration in a disordered type-II superconductor by molecular dynamics simulations of interacting vortices and find scaling laws for the front position and the density profile. The scaling can be understood by performing a coarse graining of the system and writing a disordered nonlinear diffusion equation. Integrating numerically the equation, we observe a crossover from flat to fractal front penetration as the system parameters are varied. The value of the fractal dimension indicates that the invasion process is described by gradient percolation.

Various physical, social, and biological systems generate complex fluctuations with correlations across multiple time scales. In physiologic systems, these long-range correlations are altered with disease and aging. Such correlated fluctuations in living systems have been attributed to the interaction of multiple control systems; however, the mechanisms underlying this behavior remain unknown. Here, we show that a number of distinct classes of dynamical behaviors, including correlated fluctuations characterized by 1͞f scaling of their power spectra, can emerge in networks of simple signaling units. We found that, under general conditions, complex dynamics can be generated by systems fulfilling the following two requirements, (i) a ''small-world'' topology and (ii) the presence of noise. Our findings support two notable conclusions. First, complex physiologic-like signals can be modeled with a minimal set of components; and second, systems fulfilling conditions i and ii are robust to some degree of degradation (i.e., they will still be able to generate 1͞f dynamics). C omplex systems are typically composed of interacting units that communicate information and are able to process and withstand a broad range of stresses (1-4). In physiology, freerunning healthy systems typically generate complex output signals that have long-range correlations [i.e., a 1͞f decay of the power spectra for low frequencies (**, 5-7)]. Deviations from the 1͞f pattern have been associated with disease or aging in various contexts (3,8).Despite its practical and fundamental interest (9), the origin of such correlated dynamics remains an unsolved problem (4). Until recently, attention has focused primarily on the complexity of the specific physiologic subsystems or on the nature of the nonlinear interactions between them (10-12). In particular, Boolean variables (which can take one of two values, 0 or 1) and Boolean functions have been extensively used to model the state and dynamics of complex systems (see ref. 12 for an introduction). The reason such a ''simplistic'' description may be appropriate arises from the fact that Boolean variables provide good approximations to the nonlinear functions encountered in many control systems (10,(13)(14)(15). Random Boolean networks (RBNs) were proposed by Kauffman (10) as models of genetic regulatory networks, and they have also been studied in other contexts (13,14). Wolfram (15), in contrast, proposed that cellular automata models, which are a class of ordered Boolean networks with identical units, may explain the real-word complexity. Neither of these two classes of models has been shown to generate the complex dynamics with 1͞f fluctuations observed in healthy physiologic systems.Here, we propose a modeling approach (Fig. 1a) that departs from traditional approaches in that we pay special attention to the topology of the network of interactions (4) and the role of noise (16). Our model is rooted in the following two considerations that are observed frequently in real-world systems. (i) The unit...