Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two followi
ng cases: we say $T$ is explosive if [ lim_{nto infty}m_n(T) < infty, ] and say that $T$ exhibits linear growth if [ liminf_{nto infty} frac{m_n(T)}{n} > 0. ] We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.
Robust network design refers to a class of optimization problems that occur when designing networks to efficiently handle variable demands. The notion of hierarchical hubbing was introduced (in the narrow context of a specific robust network design q
uestion), by Olver and Shepherd [2010]. Hierarchical hubbing allows for routings with a multiplicity of hubs which are connected to the terminals and to each other in a treelike fashion. Recently, Frechette et al. [2013] explored this notion much more generally, focusing on its applicability to an extension of the well-studied hose model that allows for upper bounds on individual point-to-point demands. In this paper, we consider hierarchical hubbing in the context of a previously studied (and extremely natural) generalization of the hose model, and prove that the optimal hierarchical hubbing solution can be found efficiently. This result is relevant to a recently proposed generalization of the VPN Conjecture.
