Exponential random graph models with soft clique constraints

Let r≥3 be fixed, and let Gn be the set of all simple graphs with vertex set [n]=1,ldots,n. We consider an exponential random graph model which gives higher probability to G ∈ Gn than to H ∈ Gn if G has fewer r-cliques than H. But all graphs in Gn have positive probability. The degree to which graphs with fewer r-cliques are given higher probability is determined by a positive weight w. We prove that, asymptotically almost surely as n → ∞, a random graph from Gn has a vertex partition into r-1 parts of roughly equal size, the density of edges between the parts is close to 1/2, and for every varepsilon > 0 the density of edges within any part is less than varepsilon. The asymptotic structural properties are independent of the weight w as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.