Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graph, which we call a tree of transitive tournaments. The latter is a block graph combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behaviour of a Markov random field.

We are also interested in the identifiability of the model parameters in the case when some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the ‘semi-generic’ complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of Angel et al [Random orderings and unique ergodicity of automorphism groups. J. Eur. Math. Soc., 16 (2014), 2059–2095], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in Zucker [Amenability and unique ergodicity of automorphism groups of Fraïssé structures. Fund. Math., 226 (2014), 41–61]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in Jasiński et al [Ramsey precompact expansions of homogeneous directed graphs. Electron. J. Combin., 21(4), (2014), 31] and the papers cited therein.

We analyse how a change in the probability of winning a tournament affects an agent's effort using the qualification rules for entry into the group and playoff stages of the UEFA Champions' League. Our results suggest that increasing the number of slots that a national league gets in the Champions' League leads to increases in investment in talent ex ante. This effect is largest among the teams that in the previous season just failed to qualify. This suggests that changes in prize structure leads to changes in investment decisions amongst those clubs most affected at the margin. However, we also find that incumbent teams that have already qualified for the Champions' League simultaneously raise their efforts, consistent with the occurrence of an arms race among top European football teams.

The metabolic and performance benefits of prior consumption of low-glycaemic index (GI) meals v. high-GI meals were determined in extended high-intensity intermittent exercise. Participants (ten males and four females, aged 25·8 (sd 7·3) years) completed two testing days (each consisting of back-to-back 90-min intermittent high-intensity treadmill running protocols separated by 3 h) spaced by at least 7 d. Using a randomised counterbalanced cross-over design, low-GI, lentil-based meals (GI about 42) or high-GI, potato-based meals (GI about 78) matched for energy value were consumed 2 h before, and within 1 h after, the first exercise session. Performance was measured by the distance covered during five 1-min sprints (separated by 2·5 min walking) at the end of each exercise session. Peak postprandial blood glucose was higher by 30·8 % in the high-GI trial compared with the low-GI trial, as was insulin (P = 0·039 and P = 0·003, respectively). Carbohydrate oxidation was lower by 5·5 % during the low-GI trials compared with the high-GI trials at the start of the first exercise session (P < 0·05). Blood lactate was significantly higher (6·1 v. 2·6 mmol/l; P = 0·019) and blood glucose significantly lower (4·8 v. 5·4 mmol/l; P = 0·039) at the end of the second exercise session during the high-GI trial compared with the low-GI trial. Sprint distance was not significantly different between conditions. A low-GI meal improved the metabolic profile before and during extended high-intensity intermittent exercise, but did not affect performance. Improvements in metabolic responses when consuming low-GI meals before exercise may be beneficial to the long-term health of athletes.

This paper is concerned with a random walk process in which and for i = 1, 2, ···, 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ω n defined by . The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics.

We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.