Abstract
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.
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Funding
Zhukovskii acknowledges the support of the Ministry of Science and Higher Education of the Russian Federation, Goszadanie no. 075-00337-20-03, project no. 0714-2020-0005. Zhukovskii proved Theorem 2 and derived Theorem 3 from Lemma 1. The part of the study made by Malyshkin was funded by the Russian Foundation for Basic Research, project no. 19-31-60021. Malyshkin proved Theorem 4 and Lemma 1.
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Translated by I. Ruzanova
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Zhukovskii, M.E., Malyshkin, Y.A. On the Convergence of Probabilities of First-Order Sentences for Recursive Random Graph Models. Dokl. Math. 102, 384–386 (2020). https://doi.org/10.1134/S1064562420050464
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DOI: https://doi.org/10.1134/S1064562420050464