I would like to know if there is a natural way of transforming a graph in which one seeks the shortest path between two nodes into a riemannian manifold whose this graph is a subspace. My idea is to consider the distance between two consecutive nodes of a graph $ G $ as a variable $ h(G) $ and to build a sequence of graphs $ (G_{n})_n $ such that $ G_0=G $ and for all $ i\leq j $ $ G_i $ is a subgraph of $ G_j $, with $ h(G_n) $ a decreasing function of $ n $ tending to $ 0 $ as $ n $ tends to infinity and $ G_n $ tending to a bounded domain of a riemannian manifold, its metric tensor being in some sense the limit of the weights between adjacent nodes. That way the shortest path between nodes $ A $ and $ B $ in $ G_n $ tends to the geodesic joining $ A $ and $ B $ in the limit manifold. One can process backwards and from this manifold determine the considered geodesic solving the geodesic equation and recover the shortest path in $ G_n $ for any given $ n $ as a sequence of points of this geodesic that belong to $ G_n $. In this framework, can the principle of stationary action lead to an optimal algorithm to find the shortest path in a graph ?
Has this been considered before ? If yes, could I get some reference ?