  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 graphs, which we call a tree of transitive
tournaments. The latter are block graphs 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 behavior of a Markov random field. We are
also interested in the identifiability of the model parameters in case 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.
