Showing papers by "Vito Latora published in 2000"
TL;DR: The neural system of the nematode C. elegans, the collaboration graph of film actors, and the oldest US subway system, can now be studied also as metrical networks and are shown to be small-worlds.
Abstract: The small-world phenomenon, popularly known as six degrees of separation, has been mathematically formalized by Watts and Strogatz in a study of the topological properties of a network. Small-world networks are defined in terms of two quantities: they have a high clustering coefficient C like regular lattices and a short characteristic path length L typical of random networks. Physical distances are of fundamental importance in applications to real cases; nevertheless, this basic ingredient is missing in the original formulation. Here, we introduce a new concept, the connectivity length D, that gives harmony to the whole theory. D can be evaluated on a global and on a local scale and plays in turn the role of L and 1/C. Moreover, it can be computed for any metrical network and not only for the topological cases. D has a precise meaning in terms of information propagation and describes in a unified way, both the structural and the dynamical aspects of a network: small-worlds are defined by a small global and local D, i.e., by a high efficiency in propagating information both on a local and global scale. The neural system of the nematode C. elegans, the collaboration graph of film actors, and the oldest US subway system, can now be studied also as metrical networks and are shown to be small-worlds.
300 citations
TL;DR: In this paper, it was shown that for the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate and instead, the non-extensive entropy S q ≡ (1− ∑ i=1 W p i q ) (q−1), which is a special 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.
121 citations
TL;DR: The Hamiltonian mean field model as mentioned in this paper describes a system of N fully coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased.
Abstract: We discuss recent results obtained for the Hamiltonian mean field model. The model describes a system of N fully coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased. Strong chaos is found in correspondence to the critical point on top of a weak chaotic regime which characterizes the motion at low energies. For a small region around the critical point, we find anomalous (enhanced) diffusion and Levy walks in a transient temporal regime before the system relaxes to equilibrium.
26 citations
Posted Content•
TL;DR: In this paper, the authors consider several low-dimensional chaotic maps started in far-from-equilibrium initial conditions and study the process of relaxation to equilibrium, showing that the Boltzmann-Gibbs entropy increases linearly in time with a slope equal to the Kolmogorov-Sinai entropy rate.
Abstract: We consider several low--dimensional chaotic maps started in far-from-equilibrium initial conditions and we study the process of relaxation to equilibrium. In the case of conservative maps the Boltzmann-Gibbs entropy S(t) increases linearly in time with a slope equal to the Kolmogorov-Sinai entropy rate. The same result is obtained also for a simple case of dissipative system, the logistic map, when considered in the chaotic regime. A very interesting results is found at the chaos threshold. In this case, the usual Boltzmann-Gibbs is not appropriate and in order to have a linear increase, as for the chaotic case, we need to use the generalized q-dependent Tsallis entropy $S_q(t)$ with a particular value of a q different from 1 (when q=1 the generalized entropy reduces to the Boltzmann-Gibbs). The entropic index q appears to be characteristic of the dynamical system.
25 citations
TL;DR: In this paper, the authors studied the chaotic properties of the Hamiltonian mean field model (HMF) and the Lyapunov exponents and the Kolmogorov-Sinai entropy, namely their dependence on the number of degrees of freedom and energy density.
Abstract: We study chaos in the Hamiltonian Mean Field model (HMF), a system with many degrees of freedom in which $N$ classical rotators are fully coupled. We review the most important results on the dynamics and the thermodynamics of the HMF, and in particular we focus on the chaotic properties.We study the Lyapunov exponents and the Kolmogorov--Sinai entropy, namely their dependence on the number of degrees of freedom and on energy density, both for the ferromagnetic and the antiferromagnetic case.
11 citations
TL;DR: In this article, the authors studied the chaotic properties of the Hamiltonian mean field model (HMF) and the Lyapunov exponents and the Kolmogorov-Sinai entropy, namely their dependence on the number of degrees of freedom and energy density.
Abstract: We study chaos in the Hamiltonian Mean Field model (HMF), a system with many degrees of freedom in which $N$ classical rotators are fully coupled. We review the most important results on the dynamics and the thermodynamics of the HMF, and in particular we focus on the chaotic properties.We study the Lyapunov exponents and the Kolmogorov--Sinai entropy, namely their dependence on the number of degrees of freedom and on energy density, both for the ferromagnetic and the antiferromagnetic case.
1 citations