  We consider the edge-triangle model (or Strauss model), and focus on the
asymptotic behavior of the triangle density when the size of the graph
increases to infinity. This random graph belongs to the class of exponential
random graphs, which follows the statistical mechanics approach of introducing
a Hamiltonian to weigh the probability measure on the state space of graphs. In
the analyticity region of the free energy, we prove a law of large numbers for
the triangle density. Along the critical curve, where analyticity breaks down,
we show that the triangle density concentrates with high probability in a
neighborhood of its typical value. A predominant part of our work is devoted to
the study of a mean-field approximation of the edge-triangle model, where
explicit computations are possible. In this setting we can go further, and
additionally prove a standard and non-standard central limit theorem at the
critical point, together with many concentration results obtained via large
deviations and statistical mechanics techniques. Despite a rigorous comparison
between these two models is still lacking, we believe that they are
asymptotically equivalent in many respects, therefore we formulate conjectures
on the edge-triangle model, partially supported by simulations, based on the
mean-field investigation.
