On the distribution of the transitive closure in a random acyclic digraph

Abstract

In the usual Gn,p-model of a random acyclic digraph let γ*n(1) be the she of the reflexive, transitive closure of node 1, a source node; then the distribution of γn*(1) is given by (Formula presented.) where q = 1 - p. Our analysis points out some surprising relations between this distribution and known functions of the number theory. In particular we find for the expectation of γ*n(1): (Formula presented.) where (Formula presented.) is the so-called Lambert Series, which corresponds to the generating function of the divisor-function. These resuits allow us to improve the expected running time for the computation of the transitive closure in a random acyclic digraph and in particular we can ameliorate in some cases the analysis of the Goralčíková-Koubek Algorithm.