his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_

The classical algorithms require order n ~ operations to compute the first n terms in the reversion of a power series or the composition of two series, and order nelog n operations if the fast Founer transform is used for power series multiplication In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only order (n log n) ~/2 operations In many cases of practical importance only order n log n operations are required, these include certain special functions of power series and power series solution of certain differential equations Applications to root-finding methods which use inverse interpolation and to queueing theory are described, some results on multivariate power series are stated, and several open questions are mentioned.

One of the classical topics in discrete mathematics and in computer science is the computation of the transitive closure and reduction of a directed graph. We provide a complete analysis of the random variables counting the size of the transitive closure and reduction in random acyclic digraphs. In particular we present closed forms for their probability distribution and moments, showing that they involve functions such as the q--binomial coefficients and the q--Stirling numbers of the second kind. In general it turns out that a pure numbertheoretical notation, the q--notation, represents the perfect tool to deal with these kinds of problems. In this context we speak therefore about q--distributions. 1 Introduction Graphs are probably the most common "abstract" structure in computer science and because of their flexibility they provide an adequate model for a variety of problems. Any system consisting of a set V of discrete states (called nodes or vertices) and a set E of relations b...

Predicting direct protein interactions from affinity purification mass spectrometry data