  Based on two classical notions of curvature for curves in general metric
spaces, namely the Menger and Haantjes curvatures, we introduce new definitions
of sectional, Ricci and scalar curvature for networks and their higher
dimensional counterparts. These new types of curvature, that apply to weighted
and unweighted, directed or undirected networks, are far more intuitive and
easier to compute, than other network curvatures. In particular, the proposed
curvatures based on the interpretation of Haantjes definition as geodesic
curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite
flexible. We also propose even simpler and more intuitive substitutes of the
Haantjes curvature, that allow for even faster and easier computations in
large-scale networks.
