On the convergence of probabilities of first-order sentences for recursive random graph models

We study first-order zero–one law and the first-order convergence law for two recursive random graph models, namely, the uniform and preferential attachment models. In the uniform attachment model, a new vertex with $m$ edges chosen uniformly is added at every moment, while, in the preferential attachment model, the distribution of second ends of these edges is not uniform, but rather the probabilities are proportional to the degrees of the respective vertices.