  We study the time constant $C(\nu)$ of last passage percolation on the
complete directed acyclic graph on the set of non-negative integers, where
edges have i.i.d. weights with distribution $\nu$ with support included in
$\{-\infty\}\cup\mathbb{R}$. We show that $\nu \mapsto C(\nu)$ is strictly
increasing in $\nu$. We also prove that $C(\nu)$ is continuous in $\nu$ for a
large set of measures $\nu$. Furthermore, when $\nu$ is purely atomic, we show
that $C(\nu)$ is analytic with respect to the weights of the atoms. In the
special case of two positive atoms, it is an explicit rational function of
these weights.
