Phase transition in the random triangle model

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parametersp∊ [0,1] andq≥ 1, and a finite graphG= (V,E), it assigns to elements η of {0,1}Eprobabilities which are proportional to wheret(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only ifp= (q−1)−2/3andq>qcfor a critical valueqcwhich turns out to equal It is furthermore demonstrated that phase transition cannot occur unlessp=pc(q), the critical value for percolation of open edges for givenq. This implies that forq≥qc,pc(q) = (q−1)−2/3.