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Graphical models with heavy-tailed factors can be used to model extremaldependence or causality between extreme events. In a Bayesian network,variables are recursively defined in terms of their parents according to adirected acyclic graph (DAG). We focus on max-linear graphical models withrespect to a special type of graphs, which we call a tree of transitivetournaments. The latter are block graphs combining in a tree-like structure afinite number of transitive tournaments, each of which is a DAG in which everytwo nodes are connected. We study the limit of the joint tails of themax-linear model conditionally on the event that a given variable exceeds ahigh threshold. Under a suitable condition, the limiting distribution involvesthe factorization into independent increments along the shortest trail betweentwo variables, thereby imitating the behavior of a Markov random field. We arealso interested in the identifiability of the model parameters in case somevariables are latent and only a subvector is observed. It turns out that theparameters are identifiable under a criterion on the nodes carrying the latentvariables which is easy and quick to check.

Max-linear graphical models with heavy-tailed factors on trees of transitive tournaments