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This paper provides an overview of results, concerning longest or heaviestpaths, in the area of random directed graphs on the integers along with someextensions. We study first-order asymptotics of heaviest paths allowing weightsboth on edges and vertices and assuming that weights on edges are signed. Weaim at an exposition that summarizes, simplifies, and extends proof ideas. Wealso study sparse graph asymptotics, showing convergence of the weighted randomgraphs to a certain weighted graph that can be constructed in terms of Poissonprocesses. We are motivated by numerous applications, ranging from ecology toparallel computing model. It is the latter set of applications thatnecessitates the introduction of vertex weights. Finally, we discuss some openproblems and research directions.

Limiting properties of random graph models with vertex and edge weights