Phase transition in the random triangle model

The random triangle model was recently introduced as a random graph model t hat captures the property of transitivity that is often found in social net works, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p epsilon [0, 1] and q greater than or equal to 1, and a finite graph G = (V, E), it assigns to elements eta of {0, 1}(E) probabilities which are proportional to Pi(e epsilon E)(p eta(e))(1-p)(1-eta(e))q(t(eta)), where t(eta) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangula r lattice is studied. By mapping the system onto an Ising model with extern al field on the hexagonal lattice, it is shown that phase transition occurs if and only if p (q-1)(-2/3) and q > q(c) for a critical value q(c) which turns out to equal 27 + 15 root 3 approximate to 52.98. It is furthermore d emonstrated that phase transition cannot occur unless p = p,(q), the critic al value for percolation of open edges for given q. This implies that for q greater than or equal to q(c), p(c)(q) = (q - 1)(-2/3).