  In this work we review some recent development in the mathematical modelling
of quantitative sociology by means of statistical mechanics. After a short
pedagogical introduction to static and dynamic properties of many body systems,
we develop a theory for agents interactions on random graph.
  Our approach is based on describing a social network as a graph whose nodes
represent agents and links between two of them stand for a reciprocal
interaction. Each agent has to choose among a dichotomic option (i.e. agree or
disagree) with respect to a given matter and he is driven by external
influences (as media) and peer to peer interactions. These mimic the imitative
behavior of the collectivity and may possibly be zero if the two nodes are
disconnected. For this scenario we work out both the dynamics and the
corresponding equilibria (statics). Once the 2-body theory is completely
explored, we analyze, on the same random graph, a "diffusive strategic
dynamics" with pairwise interactions, where detailed balance constraint is
relaxed. The dynamic encodes some relevant processes which are expected to play
a crucial role in the approach to equilibrium in social systems, i.e. diffusion
of information and strategic choices. We observe numerically that such a
dynamics reaches a well defined steady state that fulfills a "shift property":
the critical interaction strength for the phase transition is lower with
respect to the one expected from canonical equilibrium.
  Finally, we show how the stationary states obtained with this dynamics can be
described by statistical mechanics equilibria of a diluted p-spin model, for a
suitable non-integer real p>2. Several implications from a sociological
perspective are discussed together with some general outlooks.
