We study a discrete-time duplication-deletion random graph model and analyse its asymptotic degree distribution. The random graphs consists of disjoint cliques. In each time step either a new vertex is brought in with probability 0 < p < 1 and attached to an existing clique, chosen with probability proportional to the clique size, or all the edges of a random vertex are deleted with probability 1 − p. We prove almost sure convergence of the asymptotic degree distribution and find its exact values in terms of a hypergeometric integral, expressed in terms of the parameter p. In the regime 0 < p < 1 2 we show that the degree sequence decays exponentially at rate p 1−p , whereas it satisfies a power-law with exponent p 2p−1 if 1 2 < p < 1. At the threshold p = 1 2 the degree sequence lies between a power-law and exponential decay.

ABSTRACT. We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n ≥ 1, choose a ball from the urn uniformly at random. With probability 1/2 < p < 1, return the ball to the urn along with another ball of the same colour. With probability 1 − p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri [4,5] and Thörnblad [17]. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1 − p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p = 1).

The converse of a tournament is obtained by reversing all arcs. If a tournament is isomorphic to its converse, it is called self-converse. Eplett provided a necessary and sufficient condition for a sequence of integers to be realisable as the score sequence of a self-converse tournament. In this paper we extend this result to generalised tournaments.

ABSTRACT. The theory of tournament limits and tournament kernels (often called graphons) is developed by extending common notions for finite tournaments to this setting; in particular we study transitivity and irreducibility of limits and kernels. We prove that each tournament kernel and each tournament limit can be decomposed into a direct sum of irreducible components, with transitive components interlaced. We also show that this decomposition is essentially unique.

Given any polynomial p in C[X], we show that the set of irreducible matrices satisfying p(A) = 0 is finite. In the specific case p(X) = X 2 − nX, we count the number of irreducible matrices in this set and analyze the arising sequences and their asymptotics. Such matrices turn out to be related to generalized compositions and generalized partitions.