A statistical mechanics approach to Granovetter theory

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–Strogatz, by 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 a Hopfield model of neural networks. Once introduced, the model is investigated through graph theory (to recover Granovetter and Watts–Strogatz results) and statistical mechanics (to recover Mac-Fadden and Brock–Durlauf results). Due to the 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: shifting the patterns from [−1,+1]→[0.+1][−1,+1]→[0.+1] implies that the larger the amount of similarities among decision makers, the stronger their relative influence, and 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 à la Curie–Weiss as the one introduced by Brock and Durlauf.

► By shifting from [−1.+1]→[0,+1][−1.+1]→[0,+1] patterns of a Hopfield model, it is possible to convert frustration into ferromagnetic dilution.
► The resulting network is able to reproduce “naturally” several topological features of real social networks.
► The resulting network is able to reproduce “naturally” several econometric features of real social networks.
► We extended the Hamilton–Jacobi technique for solving the thermodynamics of this network.