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      A statistical mechanics approach to Granovetter theory

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          Abstract

          In this paper we try to bridge breakthroughs in quantitative sociology/econometrics pioneered during the last decades by Mac Fadden, Brock-Durlauf, Granovetter and Watts-Strogats through introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors. Our model relies on a pairwise Hamiltonian for decision maker interactions which naturally extends the multi-populations approaches by shifting and biasing the pattern definitions of an Hopfield model of neural networks. Once introduced, the model is investigated trough graph theory (to recover Granovetter and Watts-Strogats results) and statistical mechanics (to recover Mac-Fadden and Brock-Durlauf results). Due to internal symmetries of our model, the latter is obtained as the relaxation of a proper Markov process, allowing even to study its out of equilibrium properties. The method used to solve its equilibrium is an adaptation of the Hamilton-Jacobi technique recently introduced by Guerra in the spin glass scenario and the picture obtained is the following: just by assuming that the larger the amount of similarities among decision makers, the stronger their relative influence, this is enough to explain both the different role of strong and weak ties in the social network as well as its small world properties. As a result, imitative interaction strengths seem essentially a robust request (enough to break the gauge symmetry in the couplings), furthermore, this naturally leads to a discrete choice modelization when dealing with the external influences and to imitative behavior a la Curie-Weiss as the one introduced by Brock and Durlauf.

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          Collective dynamics of 'small-world' networks.

          Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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            The structure and function of complex networks

             M. Newman (2003)
            Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
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              Statistical mechanics of complex networks

              Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks.
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                Author and article information

                Journal
                1012.1272

                Social & Information networks, General physics

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