Andrea Rapisarda

Efficiency of scale-free networks: error and attack tolerance

The concept of network efficiency, recently proposed to characterize the properties of small-world networks, is here used to study the effects of errors and attacks on scale-free networks. Two different kinds of scale-free networks, i.e., networks with power law P(k), are considered: (1) scale-free networks with no local clustering produced by the Barabasi–Albert model and (2) scale-free networks with high clustering properties as in the model by Klemm and Eguiluz, and their properties are compared to the properties of random graphs (exponential graphs). By using as mathematical measures the global and the local efficiency we investigate the effects of errors and attacks both on the global and the local properties of the network. We show that the global efficiency is a better measure than the characteristic path length to describe the response of complex networks to external factors. We find that, at variance with random graphs, scale-free networks display, both on a global and on a local scale, a high degree of error tolerance and an extreme vulnerability to attacks. In fact, the global and the local efficiency are unaffected by the failure of some randomly chosen nodes, though they are extremely sensitive to the removal of the few nodes which play a crucial role in maintaining the networks connectivity.

Detecting complex network modularity by dynamical clustering

Based on cluster desynchronization properties of phase oscillators, we introduce an efficient method for the detection and identification of modules in complex networks. The performance of the algorithm is tested on computer generated and real-world networks whose modular structure is already known or has been studied by means of other methods. The algorithm attains a high level of precision, especially when the modular units are very mixed and hardly detectable by the other methods, with a computational effort O(KN) on a generic graph with N nodes and K links.

Vector Opinion Dynamics In A Bounded Confidence Consensus Model

We study the continuum opinion dynamics of the compromise model of Krause and Hegselmann for a community of mutually interacting agents by solving numerically a rate equation. The opinions are here represented by two-dimensional vectors with real-valued components. We study the situation starting from a uniform probability distribution for the opinion configuration and for different shapes of the confidence range. In all cases, we find that the thresholds for consensus and cluster merging either coincide with their one-dimensional counterparts, or are very close to them. The symmetry of the final opinion configuration, when more clusters survive, is determined by the shape of the opinion space. If the latter is a square, which is the case we consider, the clusters in general occupy the sites of a square lattice, although we sometimes observe interesting deviations from this general pattern, especially near the center of the opinion space.

Power-law time distribution of large earthquakes.

We study the statistical properties of time distribution of seismicity in California by means of a new method of analysis, the diffusion entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent mu=2.06+/-0.01. We propose the long-range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes.

The rate of entropy increase at the edge of chaos

Abstract Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov–Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann–Gibbs–Shannon entropy is not appropriate. Instead, the non-extensive entropy S q ≡ (1− ∑ i=1 W p i q ) (q−1) , must be used. The latter contains a parameter q, the entropic index which must be given a special value q ∗ ≠1 (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q ∗ enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

Lyapunov instability and finite size effects in a system with long range forces

Recently, the interest in phase transitions occurring infinite-size systems and the study of the related dynamicalfeatures has stimulated the investigation of the so farobscure relation between macroscopic thermodynamicalproperties and microscopic dynamical ones. In this respectseveral papers appeared in the recent literature in variousfields ranging from solid state physics [1–7] to latticefield theory [8] and nuclear physics [9,10], where thereis presently a lively debate on multifragmentation phasetransition [9–13]. The general expectation is that there isa close connection between the increase of fluctuations ata phase transition and a rapid increase of chaoticity at themicroscopic level. In several pioneering papers a differentbehavior of the largest Lyapunov exponent (LLE) l wasfound, according to the order of the transition [3,7,8,10].In particular, a well pronounced peak in LLE has beenfound for second order phase transitions, while a sharpincrease has been seen for first order phase transitions. Inthe former case some universal features have also beenfound, i.e., different systems show the same behaviorwhen properly scaled [10]. In order to connect dynamicalproperties of systems of size

Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes

We perform an analysis on the dissipative Olami-Feder-Christensen model on a small world topology considering avalanche size differences. We show that when criticality appears, the probability density functions (PDFs) for the avalanche size differences at different times have fat tails with a q-Gaussian shape. This behavior does not depend on the time interval adopted and is found also when considering energy differences between real earthquakes. Such a result can be analytically understood if the sizes (released energies) of the avalanches (earthquakes) have no correlations. Our findings support the hypothesis that a self-organized criticality mechanism with long-range interactions is at the origin of seismic events and indicate that it is not possible to predict the magnitude of the next earthquake knowing those of the previous ones.

Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system

We study the dynamics of a Hamiltonian system of N classical spins with infinite-range interaction. We present numerical results which confirm the existence of metaequilibrium quasi stationary states (QSS), characterized by non-Gaussian velocity distributions, anomalous diffusion, Levy walks and dynamical correlation in phase-space. We show that the thermodynamic limit (TL) and the infinite-time limit (ITL) do not commute. Moreover, if the TL is taken before the ITL the system does not relax to the Boltzmann–Gibbs equilibrium, but remains in this new equilibrium state where nonextensive thermodynamics seems to apply.

Nonergodicity and central-limit behavior for long-range Hamiltonians

We present a molecular dynamics test of the Central-Limit Theorem (CLT) in a paradigmatic long-range-interacting many-body classical Hamiltonian system, the HMF model. We calculate sums of velocities at equidistant times along deterministic trajectories for different sizes and energy densities. We show that, when the system is in a chaotic regime (specifically, at thermal equilibrium), ergodicity is essentially verified, and the Pdfs of the sums appear to be Gaussians, consistently with the standard CLT. When the system is, instead, only weakly chaotic (specifically, along longstanding metastable Quasi-Stationary States), nonergodicity (i.e., discrepant ensemble and time averages) is observed, and robust q-Gaussian attractors emerge, consistently with recently proved generalizations of the CLT.

Superdiffusion and Out-of-Equilibrium Chaotic Dynamics with Many Degrees of Freedoms

We study the link between relaxation to the equilibrium and anomalous superdiffusive motion in a classical N-body Hamiltonian system with long-range interaction showing a second-order phase transition in the canonical ensemble. Anomalous diffusion is observed only in a transient out-ofequilibrium regime and for a small range of energy, below the critical one. Superdiffusion is due to Levy walks of single particles and is checked independently through the second moment of the distribution, power spectra, trapping, and walking time probabilities. Diffusion becomes normal at equilibrium, after a relaxation time which diverges with N.