Many applications involve computing minimal paths over the nodes of a graph relative to a measure of pairwise node dissimilarity. These include minimal spanning trees in computer vision, shortest paths in image databases, or non-dominated anti-chains in multi-objective database search. When the nodes are random vectors and the dissimilarity is  an  increasing  function  of  Euclidean  distance  these  minimal  paths  can  have  continuum  limits  as  the  number  of nodes approaches infinity. Such continuum limits can lead to low complexity diffusion approximations to the solution of the combinatorial minimal path problem.