Barak-Erdös graphs and the infinite-bin model

Abstract : A Barak-Erd˝ os graph is a directed acyclic version of an Erd˝ os-Rényi graph. It is obtained by performing independent bond percolation with parameter p on the complete graph with vertices {1, ..., n}, in which the edge between two vertices i < j is directed from i to j. The length of the longest path in this graph grows linearly with the number of vertices, at rate C(p). In this article, we use a coupling between Barak-Erd˝ os graphs and infinite-bin models to provide explicit estimates on C(p). More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of C for p > 1/2, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of C as p → 0, using a coupling with branching random walks.
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Submitted on : Tuesday, November 28, 2017 - 3:49:40 PM
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Bastien Mallein, Sanjay Ramassamy. Barak-Erdös graphs and the infinite-bin model. 2017. ⟨ensl-01651022⟩

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