We describe an approach to the analysis of chemical (and other) networks that, in contrast to other schemes, is based on edges rather than vertices, naturally works with directed and weighted edges, extends to higher dimensional structures like simplicial complexes or hypergraphs, and can draw upon a rich body of theoretical insight from geometry. As the approach is motivated by Riemannian geometry, the crucial quantity that we work with is called Ricci curvature, although in the present setting, it is of course not a curvature in the ordinary sense, but rather quantifies the divergence properties of edges. In order to illustrate the method and its potential, we apply it to metabolic and gene co-expression networks and detect some new general features in such networks.