Preferential attachment models for randomly growing graphs have been widely studied. In the original model proposed by Barabási and Albert, each new vertex forms a link to an old vertex chosen at random, with probabilities proportional to the degrees. Bollobás, Riordan, Spencer and Tusnády proved that the degree distribution approaches a limit with a power-law tail with index $-3$, giving the scale-free behaviour seen in many real-world examples. I will talk about a family of models recently introduced by Malyshkin and Paquette, in which a new vertex first selects $r$ existing vertices preferentially, and then chooses the one with $s$th highest degree of those $r$ (breaking ties randomly). It was conjectured by Krapivsky and Redner that whenever $s>1$ the degree distribution has a doubly-exponential tail; in fact for every $s>1$ there is a transition between the conjectured behaviour and a degenerate limiting distribution, depending on the value of $r$. The only case which exhibits a third type of behaviour is when $r=2$ and $s=1$, where the tail distribution is given by a power law with index $-2$ and a logarithmic correction.
(Joint work with Jonathan Jordan (Sheffield).)

We let $W$ be a non-negative, integer-valued random variable. We consider those $W$ which satisfy a certain stochastic ordering inequality which is closely related to several well-known concepts of negative dependence. This class includes sums of negatively related indicators, and also includes ultra-log concave random variables.
Within this class of negatively dependent random variables, we may find a straightforward upper bound on the total variation distance between $W$ and a Poisson distribution with the same mean. We also have an upper bound on the Poincaré (inverse spectral gap) constant of $W$. Finally, such $W$ are smaller (in the convex sense) than a Poisson distribution of the same mean.

We consider a stochastic model of many T cells, divided into clonotypes, competing for division stimuli of many different types. Assigning a set of connections between clonotypes and stimuli produces model with competition and cross-reactivity. Relevant quantities are quasi-limiting distributions and extinction times of birth-and-death processes. Numerical solutions use the stochastic Gillespie algorithm, where each event is either the death or division of one cell. From considering distributions of times to clonal extinction, we give estimates of clonal sizes, and hence diversity, in adult mice and humans. We have in mind the homeostasis of naive human $\alpha\beta\text{CD4}^+$ T cells, but the structure of the model can be viewed as an ecological competition process involving many similar species.

(Joint work with Carmen Molina-Paris (Leeds).)