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Many complex systems can be described by networks, in which the constituent components are represented by vertices and the connections between the components are represented by edges between the corresponding vertices. A fundamental issue concerning complex networked systems is the robustness of the overall system to the failure of its constituent parts. Since the degree to which a networked system continues to function, as its component parts are degraded, typically depends on the integrity of the underlying network, the question of system robustness can be addressed by analyzing how the network structure changes as vertices are removed. Previous work has considered how the structure of complex networks change as vertices are removed uniformly at random, in decreasing order of their degree, or in decreasing order of their betweenness centrality. Here we extend these studies by investigating the effect on network structure of targeting vertices for removal based on a wider range of non-local measures of potential importance than simply degree or betweenness. We consider the effect of such targeted vertex removal on model networks with different degree distributions, clustering coefficients and assortativity coefficients, and for a variety of empirical networks.

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We present a simple two-state model illustrating the fact that the most prominent features observed in high-order harmonic generation are generic to strongly driven systems. We also address two important questions that arise in the comparison of theory and experiment.In ex riments employing very intense laser fields (I & 10' W/cm2) in gaseous media, high-order harmonics of the fundamental driving frequency have been observed. For instance, the thirty-third harmonic has been obtained in argon with 1.06 pm irradiation, and with irradiation by a 24S-nm excimer laser the seventeenth harmonic has been reported, corresponding to a wavelength of 14.6 nm. Thus far the most successful computational approach to this harmonic generation involves direct numerical integration of the time-dependent Schrodinger equation. 3 The numerical results display some of the prominent features observed experimentally:(1) the spectrum of scattered radiation consists of peaks at the odd harmonics of the fundamental driving frequency; (2) the spectrum has a plateau region in which harmonic peaks are of similar strength; and (3) there is a rapid cutoff at the highest harmonics.In this Rapid Communication we first describe a simple model for this high-order harmonic generation. Our purpose in doing so is mainly to point out that the same qualitative features (1)-(3) just mentioned are generic to strongly driven systems, and are not peculiar to atoms in quasimonochromatic fields. Furthermore, such a simple model allows us to study some aspects of harmonic generation with a minimum of computational effort.A second purpose of this paper is to address two questions that, to our knowledge, have not previously been discussed in the literature: (1) Should the calculation of the spectrum be based on the power spectrum of the expectation value of the induced dipole moment, or of the dipole correlation function? We show that it should be based on the latter. However, we also suggest why the existing theories, which employ only the dipole expectation value, may in fact be relevant to the experiments. (2) What is the appropriate power of the frequency that should multiply the square of the Fourier transform of the dipole correlation function (or expectation value)? We show that the answer depends critically on whether one is considering single-atom scattering or the actual experimental situation in which multiatom scattering and propagation effects must be addressed. We suggest, therefore, that spectra measured from single atoms or atomic beams would be different from those measured in the experiments cited above.The model we consider is quantum mechanical but involves only two atomic states. Thus we consider the following optical Bloch equations for a two-state atom in a quasimonochromatic field: -coy, y -max+ n sin(mt )z,(lb) (lc) 0 0 FREQUENCY FIG. 1. Log spectrum S(m) for a two-level model with 0/co 0.5 and m/mo 0.25.where mo is the transition frequency, z is the difference between the upper-and lower-state probabilities, and x and y involve cross pr...

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We report a study of atomic motion in time-dependent optical potentials. We measure momentum transfer in parameter regimes for which the classical dynamics are chaotic, and observe the quantum suppression of chaos by dynamical localization. The high degree of control over the experimental parameters enables detailed comparisons with theoretical predictions, and opens new avenues for investigating quantum chaos.

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We propose a method to laser-cool atoms based on stochastic cooling, first developed at CERN to cool antiprotons. Fluctuations in the momentum distribution will be detected in a pump-probe configuration with far-detuned lasers, and the appropriate correction kick will be accomplished with optical dipole potentials. Each stage of an iterative cooling process will involve measurement and feedback, with phase space remixing in between. We discuss possible applications of this method to magnetically trapped atoms and molecules.