  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.
