Abstract:
The Hamiltonian Mean Field model describes a system of N fully-coupled particles showing a second-order phase transition as a function of the energy. The dynamics of the model presents interesting features in a small energy region below the critical point. In particular, when the particles are prepared in a ``water bag'' initial state, the relaxation to equilibrium is very slow. In the transient time the system lives in a dynamical quasi-stationary state and exhibits anomalous (enhanced) diffusion and L\'evy walks. In this paper we study temperature and velocity distribution of the quasi-stationary state and we show that the lifetime of such a state increases with N. In particular when the $N\to \infty$ limit is taken before the $t \to \infty$ limit, the results obtained are different from the expected canonical predictions. This scenario seems to confirm a recent conjecture proposed by C.Tsallis.

Abstract:
We introduce a new measure of centrality, the information centrality C^I, based on the concept of efficient propagation of information over the network. C^I is defined for both valued and non-valued graphs, and applies to groups and classes as well as individuals. The new measure is illustrated and compared to the standard centrality measures by using a classic network data set.

Abstract:
The small-world phenomenon has been already the subject of a huge variety of papers, showing its appeareance in a variety of systems. However, some big holes still remain to be filled, as the commonly adopted mathematical formulation suffers from a variety of limitations, that make it unsuitable to provide a general tool of analysis for real networks, and not just for mathematical (topological) abstractions. In this paper we show where the major problems arise, and how there is therefore the need for a new reformulation of the small-world concept. Together with an analysis of the variables involved, we then propose a new theory of small-world networks based on two leading concepts: efficiency and cost. Efficiency measures how well information propagates over the network, and cost measures how expensive it is to build a network. The combination of these factors leads us to introduce the concept of {\em economic small worlds}, that formalizes the idea of networks that are "cheap" to build, and nevertheless efficient in propagating information, both at global and local scale. This new concept is shown to overcome all the limitations proper of the so-far commonly adopted formulation, and to provide an adequate tool to quantitatively analyze the behaviour of complex networks in the real world. Various complex systems are analyzed, ranging from the realm of neural networks, to social sciences, to communication and transportation networks. In each case, economic small worlds are found. Moreover, using the economic small-world framework, the construction principles of these networks can be quantitatively analyzed and compared, giving good insights on how efficiency and economy principles combine up to shape all these systems.

Abstract:
This paper elucidates the connection between the KS entropy-rate kappa and the time evolution of the physical or statistical entropy S. For a large family of chaotic conservative dynamical systems including the simplest ones, the evolution of S(t) for far-from-equilibrium processes includes a stage during which S is a simple linear function of time whose slope is kappa. The paper presents numerical confirmation of this connection for a number of chaotic symplectic maps, ranging from the simplest 2-dimensional ones to a 4-dimensional and strongly nonlinear map.

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-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.

Abstract:
Critical infrastructure networks are a key ingredient of modern society. We discuss a general method to spot the critical components of a critical infrastructure network, i.e. the nodes and the links fundamental to the perfect functioning of the network. Such nodes, and not the most connected ones, are the targets to protect from terrorist attacks. The method, used as an improvement analysis, can also help to better shape a planned expansion of the network.

Abstract:
We introduce the concept of efficiency of a network, measuring how efficiently it exchanges information. By using this simple measure small-world networks are seen as systems that are both globally and locally efficient. This allows to give a clear physical meaning to the concept of small-world, and also to perform a precise quantitative a nalysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency.

Abstract:
The mathematical study of the small-world concept has fostered quite some interest, showing that small-world features can be identified for some abstract classes of networks. However, passing to real complex systems, as for instance transportation networks, shows a number of new problems that make current analysis impossible. In this paper we show how a more refined kind of analysis, relying on transportation efficiency, can in fact be used to overcome such problems, and to give precious insights on the general characteristics of real transportation networks, eventually providing a picture where the small-world comes back as underlying construction principle.

Abstract:
The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multi-layer networks, i.e. networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here, we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real-world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers, and between activities and degrees. We show that real-world networks exhibit indeed non-trivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.