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. ...

We investigate fracture and fragmentation of solids due to impact at low energies using a twodimensional dynamical model of granular solids. Simulating collisions of two solid discs we show that, depending on the initial energy, the outcome of a collision process can be classified into two states: a damaged and a fragmented state with a sharp transition in between. We give numerical evidence that the transition point between the two states behaves as a critical point, and we discuss the possible mechanism of the transition. PACS number(s): 64.60.Ak, 46.30.Nz

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Here we present a model that is based on self-organized criticality and takes into account brain plasticity, which is able to reproduce the spectrum of electroencephalograms (EEG). The model consists of an electrical network with threshold firing and activity-dependent synapse strengths. The system exhibits an avalanche activity in a power-law distribution. The analysis of the power spectra of the electrical signal reproduces very robustly the power-law behavior with the exponent 0.8, experimentally measured in EEG spectra. The same value of the exponent is found on small-world lattices and for leaky neurons, indicating that universality holds for a wide class of brain models.

Résumé. 2014 Breaking phenomena of disordered structures such as the tearing of unwoven textiles, the fracture of brittle materials, or the propagation of cracks in solids are receiving increasing attention due to their imminent technological relevance and due to the fundamental theoretical questions involved [1][2][3][4][5][6][7]. The irreversibility and non-locality of breaking processes produces rather complicated spatial and temporal correlations, giving rise to a potentially much richer phenomenology than the extensively studied steady-state properties of disordered media, such as the electrical conductivity or superconductivity [8][9][10][11], or the linear elasticity of random networks [12][13][14]. There is a vast literature on the general problem of mechanical failure and crack propagation in solids [1, 6,7]. Unfortunately much of the existing work involves very complicated models and/or calculations. Our motivation is to introduce a relatively simple and tractable model which captures at least some of the basic features of breaking processes. To this end, we consider the electrical analogue of breaking through the introduction of a random fuse network.(*) Supported in part by grants from the ARO, NSF, and ONR.Article published online by EDP Sciences and available at http://dx