Convergence in homogeneous random graphs

For a sequence p̄ = (p(1), p(2), ‥ .), let G(n, p̄) denote the random graph with vertex set {1, 2, ‥ ., n} in which two vertices i, j are adjacent with probability p(|i − j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n, p̄), can be affected by the behaviour of p̄ and the strength of the language we use.