Harmony in the Small-World

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-worlds 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 the 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 term of information propagation and describes in an 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 on a global scale. The neural system of the nematode C. elegans, the collaboration graph of film actors, and the oldest U.S. subway system, can now be studied also as metrical networks and are shown to be small-worlds.Comment: 16 pages, 3 figures, accepted for publication in Physica

The Architecture of Complex Systems

A short review of the recent results and models of complex networks.Comment: 8 pages, 1 figure. To appear on "Interdisciplinary Applications of Ideas from Nonextensive Statistical Mechanics and Thermodynamics", Santa Fe Institute for Studies of Complexity. Oxford University Pres

Compromise and Synchronization in Opinion Dynamics

We discuss two models of opinion dynamics. First wepresent a brief review of the Hegselmann and Krause (HK) compromise model in two dimensions, showing that it is possible to simulate the dynamics in the limit of an infinite number of agents by solving numerically a rate equation for a continuum distribution of opinions. Then, we discuss the Opinion Changing Rate (OCR) model, which allows to study under which conditions a group of agents with a different natural tendency (rate) to change opinion can find the agreement. In the context of the this model, consensus is viewed as a synchronization process.Comment: Talk presented at the international conference Next05 Sigma Phi, 13-18 august 2005, Kolymbari, Crete. EPJ B (2006) in press. Typos corrected, refs adde

Dynamics and Thermodynamics of a model with long-range interactions

The dynamics and the thermodynamics of particles/spins interacting via long-range forces display several unusual features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular the model shows a second order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact the numerical simulations show the presence of quasi-stationary state (QSS), i.e. metastable states which become stable if the thermodynamic limit is taken before the infinite time limit. The QSS differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heat, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly-decaying correlations and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS have been recently interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which is not alternative to the previous one.Comment: 12 pages, 5 figures. Recent review paper for Continuum Mechanics and Thermodynamic

Multiple centrality assessment in Parma : a network analysis of paths and open spaces

One of the largest of Europe, the recently realized university campus 'Area of the Sciences' in Parma, northern Italy, has been planned for a comprehensive programme of renovation and revitalization with a special focus on vehicular accessibility and the quality of open spaces. As part of the problem setting phase, the authors, with Rivi Engineering, applied Multiple Centrality Assessment (MCA) - a process of network analysis based on primal graphs, a set of different centrality indices and the metric computation of distances - in order to understand why the existent system of open spaces and pedestrian paths is so scarcely experienced by students as well as faculty and staff members and why it appears so poorly supportive of social life and human exchange. In the problem-solving phase MCA was also applied, turning out to offer a relevant contribution to the comparative evaluation of two alternative proposed scenarios, leading to the identification of one final solution of urban design. In the present paper, the first professional application of MCA, an innovative approach to the network analysis of geographic complex systems, is presented and its relevance in the context of a problem of urban design illustrated