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To each edge (i,j), i<j of the complete directed graph on the integers weassign unit weight with probability p or weight x with probability 1-p,independently from edge to edge, and give to each path weight equal to the sumof its edge weights. If W^x_{0,n} is the maximum weight of all paths from 0 ton then W^x_{0,n}/n \to C_p(x), as n\to\infty, almost surely, where C_p(x) ispositive and deterministic. We study C_p(x) as a function of x, for fixed 0<p<1and show that it is a strictly increasing convex function that is notdifferentiable if and only if x is a nonpositive rational or a positive integerexcept 1 or the reciprocal of it. We allow x to be any real number, evennegative, or, possibly, -\infty. The case x=-\infty corresponds to thewell-studied directed version of the Erd"os-R'enyi random graph (known asBarak-Erd"os graph) for which C_p(-\infty) = lim_{x\to -\infty} C_p(x) has beenstudied as a function of p in a number of papers.

Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph